I have seen Berdan primed cases, in the past, with the stem removed, then a cup that will hold a standard Large Rifle primer soldered in to the hole. The original hole might have been enlarged slightly.

If you drive a slug through the choke, I take it the major (largest) diameter is .775". What is the measurement from groove to groove. That is the bore size.
The proofing marks may be out-of-sinc with that that gun is now.
12 bore would be .729" to .730", yes.
11 bore is .751"
10 bore is .775"

You said you tried small pistol primers and that they fit snugly? I have not seen such small Berdan primers before - of course there must be for small ctgs. just that I have never seen them on large cases.