Suppose I have a bunch of hardware sitting around, a base/CORS reference, etc. Is there an accepted best-practice way to pipe it into my phone to let this app use it? Or to take logs in my backpack and post-align the data when I get home?
Your app could interface with more accurate external sensors, and then set the phone's GPS fix to be what the sensors read. Assuming that the issue is lack of accuracy at the sensor level and not lack of precision at the software / API level.
But are you sure the GPS in your phone isn't good enough to map the location of benches? I've seen phones get within a meter or so, under good conditions - more than sufficient to get someone close enough to see a bench.
I'm sitting on a couple Ublox F9P's, one of which I need to get back in service as a Galmon monitor station but which also serves as my RTK base, and the other is on a quad that I'm not flying anymore so that's my rover.
Maps basically lie all the time and every map is an imperfect approximation of reality. Of course if your envisioned data consumers are blind people that are guided by surveying GPS equipment to that bench so they can just sit and be confident in actually sitting on the bench ... well, yeah, that might work. But I don't think that's a particular compelling use case and most people when they're looking for the bench that's on their map will find it just fine if it's not that accurate.
Interesting tidbit: Hiking guideposts around here all have UTM coordinates written on them and a few of them are a bit wrong, at least compared with other geometry sources that should in theory also be accurate (like cadastre data).