So I decided to try to spray paint it using Rustoleum's Universal paint in "Oil-Rubbed Bronze". It says it's good for all surfaces, so I hoped it would work well for something that requires flexibility. First I covered the metal parts of the belt with painter's tape. I was careful to work the tape around the edges of the studs using my fingernail, then used an Exact-o knife to cut around the stud and remove the excess tape.
I sprayed two light coats of paint, allowing a little time to dry in between. Then I let it dry completely (so it's not tacky anymore) before removing the tape.
I carefully removed the tape, and here's the finished product!
Ahh, much better. And so far there seems to be no problem with the flexing of the belt, and the paint doesn't stick to the buckle or loop. Success!
Now, what other accessories can I spray paint?.....