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.