Einstein showed in various papers that the energy content of a body was a measure of its inertial mass. If you increase the internal energy of a system while keeping the center of energy at rest, its effective inertial mass increases because of the increase of energy required to move the center of energy from rest to a velocity $v\ll c$.
Is it possible to come up with a similar argument, just using special relativity, to show the equivalence of inertial and gravitational mass?
No -- the essential problem is that there is no way to, a priori, introduce a notion of "this much mass gives you this much gravity." You have to posit some sort of law for it. Once you start requiring that this law reduces to Newtonian gravity and doesn't violate special relativity, you are pretty much forced down the road toward General Relativity.
No. The equivalence of inertial and gravitational mass is simply an experimental fact. Actually, part of what makes General Relativity a great theory is that its entire consistency rests that one experimental fact. To falsify it, all you need to do is to find an object with different inertial and gravitational masses. Einstein had a knack for coming up with theories that are both incredibly correct and incredibly easy to disprove. Special relativity is completely dependent on the lack of existence of aether, which is also just a simple empirical fact.
There are many possible ways of stating the equivalence principle, which are all logically related but not exactly equivalent.[Sotiriou 2007] One way of stating the e.p. is that GR becomes SR on small scales. From this statement of the e.p., I think it should be pretty clear that SR can't be used to prove the e.p. If principle A says that theory B is a good approximation under certain conditions, then clearly theory B can't imply principle A.
Sotiriou et al, "Theory of gravitation theories: a no-progress report," 2007, http://arxiv.org/abs/0707.2748
The effective inertial mass as you say is dependent of its direction, the angle between force and velocity. If permitted to coin temporal terms, they can be called as parallel mass, traverse mass, and slant mass.The effective inertial mass is not a single value as a rest mass.
In common physics terms "mass" means a rest mass, an inertial mass measured by the observer who moves along with the body. The rest mass is a single value independent from the angle of force (its velocity is zero).
After all special relativity is laws about the nature of space and time. It has nothing to do with gravitation.
No, you cannot, it's not possible.
