This carved keystone is on a local building. The building is entering the early stages of restoration and there are plans to build a professional theater company here. I've been hired to design the visual identity for the theater that will be called the 1871 Theater.

To me this looks like a mixing of type styles. The 8 and the 7 look like they could plausibly be a didone or fat face (though the 7 is unique - have looked through many old type specimen books and haven't found anything quite like it) while the 1s look like a Clarendon.

My hope is to anchor my lettering for the logotype(s) in historically accurate inspiration and not look like a tool (to more legit type nerds than myself). What do you think? Are these a 155 year old mashup? Has anyone seen a typeface that all these numerals could be based on?

This article by Ralf Herrmann (who runs this site) explains why what you posted has little to do with typefaces used for printing. Go to the section on Stonemasonry.

For something modern but with late 19th-century roots and a similar vibe, try Apothicaire by Sudtipos.