Is gravity Lorentz-invariant? If I go whizzing past the Sun in my spaceship at 0.8c, I will see that the Sun looks like it has been squished. That is, it will be Lorentz-contracted along the direction of my motion (by a factor of 0.6). Since there is no such thing as absolute motion, I am at liberty to regard myself as at rest, and the Sun as moving. Certainly, if the Sun were at rest with respect to me, I would expect it to be (nearly) spherical, and I would be at a loss to explain its being an oblate spheroid instead. Is there some aspect of the theory of gravitation, either Newtonian or Einsteinian, by which an object's gravitational field is modified by its motion, so as to produce this effect?
In Chapter 6 of EM Fields and Waves, by Lorrain and Corson, it is demonstrated that when the Lorentz transformation is applied to the force between two charges, considered in a frame where they are at rest vs one where they are comoving, the resulting modification of the force gives rise to terms that are mathematically identical to that which results from the magnetic field induced by the motion. In their words; "..the magnetic field in frame 1 has appeared as a result of the application of a relativistic transformation to the electric force in frame 2." That is, an observer in the moving frame would find the behavior of the charges inexplicable if he did not include magnetism in his theory of EM forces. Thus, magnetic forces are necessary to make electromagnetism Lorentz invariant. Since I am not aware of any aspect of gravitational theory which corresponds to magnetism, I do not see how gravitational theory can account for the effects of motion.     
 A: 
Since I am not aware of any aspect of gravitational theory which corresponds to magnetism, I do not see how gravitational theory can account for the effects of motion.

Gravitomagnetism is in fact a known and measured phenomenon.  It emerges from Einstein's general relativity, rather than Newtonian gravity (which, as has been noted in other answers, is definitely not Lorentz-invariant.)  The derivation itself is rather involved, but basically, one makes the following assumptions:


*

*The metric is very close to flat, i.e., $g_{ab} = \eta_{ab} + \gamma_{ab}$, where $\gamma_{ab}$ is "small"; 

*The matter sources that are responsible for the curvature are moving slowly relative to light;  and

*The test particles that are accelerating due to this curvature are moving slowly relative to light.  
(All statements above concerning velocities and accelerations are implicitly made with respect to the reference frame used to construct the flat background metric.)  You then figure out the equations obeyed by the components of $\gamma_{ab}$ given the known distribution of the source's energy density and energy flux.  Finally, you use the Christoffel symbols for this perturbed metric to look at the acceleration of a test particle moving under the gravitational influence of this source.
Under the above assumptions, you do in fact get a magnetic analog of the gravitational force!  The Newtonian force on the test particle is the term that you obtain when you set the two velocities (source and test particle) to zero.  The term that's linear in the source & test particle velocities, meanwhile, looks for all the world like a magnetic Lorentz force.  It's parallel to $\vec{v} \times \vec{B}$, where $\vec{v}$ is the test particle velocity and $\vec{B}$ depends on the energy flux distribution of the source in a way that looks for all the world like the Biot-Savart Law.
These gravitomagnetic effects do, however, lead to measurable phenomena, of which the best-known is frame-dragging.  The idea here is that the rotation of a central body will cause orbiting bodies to behave differently than if the body were not rotating, just as a configuration of moving charges will exert different forces on a test charge than if the same configuration of charges was stationary.  This effect was finally confirmed a few years ago (after many, many years of trials and tribulations) by the Gravity Probe B experiment.  The effect they were looking for was basically the gravitational analog of two magnetic dipoles exerting a torque on one another;  in their case, one of the "dipoles" was a gyroscope on the satellite, and the other was the Earth.
Unfortunately, I am unaware of any good sources that go through this derivation explicitly (if anyone puts one in the comments, I'll revise this sentence.)  Wald's General Relativity discusses the phenomenon briefly in Section 4.4, but he relegates most of the calculation to an end-of-chapter problem.  
A: Yes.
General Relativity is perfectly capable of showing how a large oblate spheroid of gas moving in a particular direction contracts to form an oblate spheroid shaped star moving in that same direction that shines and emits light.
And it is perfectly capable of making models that predict all the observations that any observer would make.
Frankly, its a bit hard to imagine that you'd think otherwise. The physics doesn't depend on your frame. So you make the same predictions for what every observer detects regardless of any choice of frame that may or may not be used to describe it. So if you can describe the collapse process in the frame of the star then you can make a model of the collapse process. And that model can be used for any observer.
The model is a geometric structure that has everything you need to make predictions. Here is an easier example. In the center of momentum frame an elastic collision has two particles come in and each reverse there velocity and then head out. That is a description in a frame, and it is particularly easy. But the model of such a process is a geometric structure. 
The geometric structure has the tangent to the worldline be proportional to the energy-momentum four vector.  So each incoming particle has its own worldline, its own unit tangent, and its own energy-momentum vector (call them p1 and p2). And at the point of collision the sum of those energy momentum vectors exists, call it s.
To find the outgoing tangent, you take the incoming tangent v1 and rotate it into s then repeat that rotation to get v1'. And similarly you take v2 and rotate it into s then repeat that rotation to get v2'.
This description is about lines and curves and tangents and such and so doesn't depend on what frame you pick or even whether you pick a frame. So we can literally predict what happens during an elastic collision without having to pick some frame or talk about who is an observer. Because we just make a geometric model of the events 
Sure, in the center of momentum inertial frame it looks like a simple bounce where each has its velocity sent to its negative, but the geometric description never required you to pick a frame.
Picking a frame is a mere convenience, like picking an x axis or a y axis. The vector equations hold and work and don't care which frame or even whether you pick a frame.
Stars cause curvature outside themselves because the kinds of vacuum curvature that was outside the star has a natural dynamic (governed by the speed of light) that fills itself into any vacuum region as the collapse leaves more vacuum outside. So that is how the spacetime outside the gas cloud becomes curved as the gas contracts and becomes the star. And General Relativity describes this fine, without ever having to pick a frame.
So there is no issue. General Relativity is a geometric theory that doesn't require that you pick a frame. It just requires that its models satisfy Einstein's Equation.

if the Sun were shaped by electrical forces, a person who observed it from a frame with large relative velocity would note that it looked squished. 

It wouldn't just looked squished. The shape and even volume of extended objects (and even whether it is a single connected object) depends on your frame.

He would attribute this departure from spherical symmetry to the magnetic field generated by the motion he perceives.

That's a gross oversimplification. The person would note that it formed into the shape it was based on the past condition (past shape and past positions and past motions and past fields) and the dynamical laws. Since they disagreed about the shapes in the past, they end up disagreeing about the shapes now.
They agree on the dynamics, for instance on the proper accelerations felt at any specific event (they agree on the rate momentum changed per proper time, they agree on the worldline of the particle).
Disagreements consist of whether a worldline's tangent vector is "at rest" or not. And whether the propoer time agrees with an inertial time. And whether an electromagnetic field has a particular amount electric field or magnetic field. But they agree on the force.
But gravity isn't a force. So people don't take 
Gravitoelectromagnetism seriously as anything other than an approximation for some situations. So they don't worry about breaking a Gravitoelectromagnetic force into a Gravitoelectric force and a Gravitomagnetic force.
The sun is a squished moving star because it formed from the collapse of a squished moving ball of gas. And when you use GR to model the collapse of a squished moving ball of gas you get exactly what you expect.
Please don't pretend this is not known and that this isn't 100% exactly what GR predicts. Even collapsing dust would do this when there are zero forces whatsoever. A squished moving ball of dust collapses into a denser squished moving ball of dust, according to regular GR, with no forces.
A: 
Is there some aspect of the theory of gravitation, either Newtonian 

Newtonian gravitation has instantaneous transmission of the gravitational field

or Einsteinian, by which an object's gravitational field is modified by its motion, so as to produce this effect? 

In general relativity  Lorenz transformations are inherent at the flat limit, see this set of question and answers; and the Newtonian gravitational field is an emergent one, from the curvature of space induced by the masses.
One needs to go to the flat space approximation and use Lorenz transformations for a field emergent from the general relativity curvature, where the gravitational waves travel also with velocity c, to explain the particular effective forces at the spaceship. Look at how the GPS system uses general relativity.
