That is fine, that is exactly what I do. The forest service allows dumping of grey water, just not black (duh).
I think it will depend on where you are and what your grey water consists of. Depending on what we've been washing off our dishes, it's possible to get some pretty stinky, fly-attracting grey water. But by using biodegradable soaps, we avoid the sudsy, high-phosphate residue that hangs around from normal soaps.
Your proposed use is a good one, though. It seems a different issue than just letting the groundwater run onto the ground at the rear of your vehicle, which can get ugly. And while lots of state and local parks, and almost all private parks, restrict grey water dumping to the provided sumps, I wouldn't think you'd be hassled for using relatively clear grey water in the firepit.