I don’t see how web clients could fetch offer invoices though, would have to use lnsocket/lnmessage and websocket proxies. With lnurl it’s much easier for web apps to fetch invoices.
Better late than never. See the Proof of Payment section at https://github.com/bitcoin/bips/pull/1555/files
Discussion
The web client shouldn’t be fetching the invoice, it just passes the whole offer on to the wallet and asks it to pay. When the wallet completes it does a callback to the initiating app.
Admittedly I wasn’t really thinking about web apps in the above design though. Nothing specific shouldn’t work except that the spec explicitly says wallets can’t open a callback URI that has a scheme of http(s). I was worried about it being a way for a payee to discover the payers IP, but I think maybe it doesn’t matter.