Hi Ben,
I can't say for sure, but I know that I would get the US passport and then BRP corrected first before going any further. Does her birth certificate and social security stuff have her full name or abbreviated? It feels like it should be OK, but there are rules about the issuance, and it might be considerably more difficult in correcting a naturalisation certificate later on.
I don't think you need to mention the BRP in that section, but I doubt it would harm it if you did. For the record, we did ours in November 2020 and the final list of documents was marriage certificate, BRP (as part of its own requirement), passports and referee declarations.
We didn't send the passports until it came time to apply for the UK passport, at which point it was away for a couple of weeks before being returned with the naturalisation certificate (which did get creased, unfortunately, so be prepared for that).