If the "365 B" is engraved and not stamped, then it appears you have recoverd a butt plate from a Confederate Enfield.
Some of the early Enfields that the Confederates imported from England were engraved with so called lot numbers on their butt plate tangs. Circumstantial evidence indicates that the numbering system went from 1 to 9999, and then continued to a second set of 9,999 similary marked Enfields having the suffix "A" below the engraved number. The next 9999 had the suffix "B" engraved below the number. To my knowledge no engraved numbers have been found beyond the "B" suffix range.
I recently saw one of these engraved butt plate tangs on a dealer's website priced at nearly $600.
Sounds like you made quite a find. These are rare.