Chuck, I have a 55T Mod 1 gear. Its OD is 56.8mm (should be N+2 = 57mm, but it's Chinese). 57mm is 2.244" , which is quite a bit less than your 2.3125", though you are relying on someone else's measurements, probably done with a rule.
If you have a mini-lathe, its changewheels will be Mod 1. Can you try one of them against the broken gear to see how it meshes?
Andy