I have a very simple scenario for you, which should be quite instructive, and should illuminate things a little.
Imagine a particle travelling along some geodesic which crosses through a given event; this shall be our reference. Now we imagine a second particle, and say that it initially travels parallel to the reference particle with some 4-velocity
u through empty space. We assume for simplicity's sake that the particles are uncharged, and don't interact except through gravitation; let's further assume they are massive particles.
Now, so long as they are travelling through empty space their geodesics will remain parallel, no surprises here. If, however, we place a massive body somewhere in the vicinity of our particles ( let's imagine it to be point-like ), that parallelism will be disturbed, and the geodesics will deviate. In standard GR this is an almost trivially simple problem, since the geodesics follow the curvature of space-time; if the separation between the two particles' geodesics is denoted by the 4-vector
v, the resulting deviation is
$$\displaystyle{\frac{D^2v}{d\lambda ^2}=-R(u,v,u)}$$
wherein R is the Riemann curvature tensor, and the capital D denotes the covariant derivative as opposed to the ordinary partial derivative. That's it. No assumptions about speeds of light, wave nature of matter etc etc needed. We are in fact not at all interested in any speeds, we only want to look at the separation between their trajectories, and separation ceases to be constant simply because space-time is no longer flat if we place a massive body in it. This is just a very simple model of gravitational attraction.
So now, how would that very simple scenario play out in your little world where space-time is flat, and only the speed of light varies ? What happens to the particles' trajectories, and why ? Remember also that we are
not assuming photons here, these could be any type of particle, we place no constraints on it.
And don't even
think about bringing out the "hiding behind the maths" argument - this is a simple, perfectly reasonable question about a perfectly simple scenario, to which we can expect an answer from you.
This should be interesting