There are two effects you need to consider.
One is that gravity depends on the full energy-momentum-stress tensor. Usually for slow moving objects the biggest component of this is the energy, all the others are negligible, and so we commonly say gravity depends on the mass (=energy) alone. But energy and momentum are just individidual coordinates of an invariant 4-dimensional quantity, and if you change to a moving reference frame it's just like rotating your coordinate system (only it is a hyperbolic rotation, it preserves the length $x^2-y^2$ instead of $x^2+y^2$, meaning that as one coordinate increases so does the other). The invariant magnitude of the 4-dimensional quantity is the energy squared minus the momentum squared. If you change to a moving coordinate system, momentum increases along with energy in synchrony, leaving the magnitude the same.
This is the same answer as to the question: when you accelerate and see the whole universe start moving past you, where does the kinetic energy of the whole universe come from? The answer is that energy and momentum are complementary coordinates of a 4-dimensional quantity, which is what is conserved, and the increase in energy is cancelled by the increase in momentum. Or more accurately, you are just rotating your coordinate system, and some of the enormous rest energy of the whole universe is appearing in the spatial direction as momentum. A lot of apparent paradoxes can be resolved because of this sort of compensatory-coordinates effect.
The other effect you need to consider is time dilation. The convergence of your moving celestial bodies can be considered to be a clock - they get so many metres closer in such and such a time, they orbit around each other with such and such a period - and relativistic velocity slows this down. The acceleration you measure in the moving frame will be different.