⟨06Starscraper⟩ ⟪egg⟫ And baseless speculation is not how you […] ⮪ It's not baseless, but there's nothing that does not begin, at first, as speculation. And I must respectfully state that no, I will not stop being sure of things. Especially not when used as a figure of speech that is motivation for researching something more deeply.
(It's not baseless because typically, you can exchange modelling accuracy for a model that's simpler to work with, and given the game application here, a fair bit of accuracy can be sacrificed. Newtonian gravity itself can be seen as a more tractable form of GR that is less precise - ie, GR reduces to newtonian gravity with certain assumptions)
As to the Einstein–Infeld–Hoffmann equations - perhaps they don't lead to frame dragging, but frame dragging was first predicted in a weak-field approximation - ie, linearized GR, in the case of Lense–Thirring precession. https://en.wikipedia.org/wiki/Lense%E2%80%93Thirring_precession
And Post-Newtonian expansions were _how_ the precession of Mercury's orbit was calculated. EIH was just the first of the whole field of Post-Newtonian Expansions that's matured a lot over the last century. 
I'm not saying it would be easy or even doable - these levels of ODE/DAE/PDE-solving are beyond me (not that they're beyond what I might be willing to try), and obviously you know how Principia is written way better than I do - but I do know there's lots of approximation methods out there, many even within linearized/weak-field GR - that lead to interesting effects being modeled, without having to confront the full complexity of GR directly.
Also, being that the goal is an interesting game experience, there are approximations I'd be willing to make that would be unacceptable for most scientific applications - as long as interesting things like frame dragging and Shapiro Delay are emulated.