Quote by DrStupid
In GR the stress–energy tensor is the source of gravitation and it is not frameinvariant.

If you mean "not frameinvariant" in the sense that it's not a scalar, true. But it is certainly "frameinvariant" in the sense that it's a tensor and transforms accordingly when you change coordinates, so contracting it with other tensors yields frameinvariant scalars. Any actual observable that tells you about the "amount of gravity produced" by an object will be such a scalar, so it will be frameinvariant, as I said.
Quote by atyy
So could one instead say that this is a case where the relativistic mass (which is just another name for energy) clarifies, indicating that nontidal gravity can be transformed away, consistent with the Principle of Equivalence?

No. The "amount of gravity produced" by the object is not a function of its energy, it's a function of its stressenergy tensor, of which energy is only one component. In a frame in which the object is moving, there will be other nonzero components of the SET as well as the energy, and their effects will offset the apparent "effect" of the increase in energy, so the final result will be the same as it is for a frame in which the object is at rest.
Quote by atyy
The main problem seems not the coordinate variance of the description, rather the question seems to assume some sort of superposition, which may not hold in GR because it is a nonlinear theory.

This is true; so far I've only talked about a single object. Since GR is nonlinear, two solutions do not add up to another solution, so I can't just take the solution for each body in isolation and add them to get a solution for both bodies. That's why we don't have a closed form solution for, e.g., binary pulsars, but have to calculate their expected orbital changes due to the radiation of gravitational waves numerically.
However, we don't have to get into that to resolve the question of whether an object being in motion changes the gravity it produces. It doesn't.
Quote by Naty1
Any situation where you ask about a rapidly moving massive body's gravitational effect and a 'stationary' observer can be transformed to an equivalent question about the interaction between a rapidly moving observer a 'stationary' massive body. So all observations relating to a rapidly moving massive body can be answered as if the body is stationary

This is key: this is almost always the quickest method of figuring out how much gravity an object produces. Transform to its rest frame, in which the stressenergy tensor will usually have its simplest form. Actually, in the case given in the OP, it's even simpler, since each object is isolated so the individual solution for the gravity produced by the object, in the vacuum region outside the object, is just the Schwarzschild solution with the object's mass M. Transforming that solution to a frame in which M is moving does not change any of its physical predictions, it just makes it look more complicated while still giving the same answers.