Sadly it’s a bit more complicated. To avoid making all lightning nodes speak nostr, you probably just want the PoP scheme, which is (a) enough to prove the BOLT 12 invoice came from the public BOLT 12 offer in the profile, (b) the payment preimage (and matching hash in the invoice), (c) the payer key signing the nostr zap info (and matching payer pubkey in the invoice). There’s a corresponding new PoP field in BIP 321 with the intention of being able to just paste/open a BIP 321 with a BOLT 12 offer (and nostr metadata requesting the PoP) and then once the payment completes the wallet can automatically jump back to the nostr client with the PoP provided.