I think the 16-gauge Model 12s were built for 2-9/16-inch chambers prior to 1926. After that, they (and pretty much everyone not determinably British) standardized on 2-3/4-inch shells. If yours is 2-3/4-inch, that'll be marked on the barrel. If there's no chamber-length marked, it should be a 2-9/16-inch chamber. Lots of these old chambers got opened up to 2-3/4-inch without trouble, but as Mr. Bear Foot Farm says, the firing pin problem should have nothing to do with the chamber length.
I doubt you can find 2-9/16-inch ammunition. I find it hard enough locating standard-chamber 16-gauge shells in these standardized times.
I doubt you can find 2-9/16-inch ammunition. I find it hard enough locating standard-chamber 16-gauge shells in these standardized times.