Now I'm wondering how chimps would respond to Newcomb's problem.

I hope, we never find out how chimps discuss the last paragraph:

... Sometimes, at least in humans, social interactions can also increase our irrationality instead. But chimps don’t seem to have this problem. Engelmann’s team is currently running a study focused on whether the choices chimps make are influenced by the choices of their fellow chimps. “The chimps only followed the other chimp’s decision when the other chimp had better evidence,” Engelmann says. “In this sense, chimps seem to be more rational than humans.”