Unfortunately, approximations that require that the speed be small (for instance the PPN formalism, i.e. Paramterized Post Newtonian approximations) can't really address the issue of how speed affects gravity, because the speed has already been assumed to be small.
The biggest pitfall to watch out for in trying to treat gravity as a "force" is how it behaves in different reference frames. I'm not aware of anything that plays the role of "force" in full GR that transforms like a tensor (much less like a 4-vector, which is what one expects a force to transform like).
The closest candidate I'm aware of are the Christoffel symbols - which are not tensors, and have the wrong rank as well.
This argument may sound technical, and it is - but not transforming properly ultimately leads to a lot of confusion. It breaks the usual way in which physics summarizes all the different things that different observers might measure into a single, unified, observer-independent framework.
In fact, I think that trying to understand how gravity transforms between different frames , how it appears to different observers, may be the root of the OP's question.