It occurred to me I don't even know if this sort of thing is allowed. You'd think it would be, though. It looks pretty restricted in Chrome (must specify a static list of domains it works with) at least via cursory glance.
Discussion
I started looking into implementing this (It does look like WebExtensions *do* support this, you just need a content script to proxy between the actual page and the background page), but it seems that nostr:npub1getal6ykt05fsz5nqu4uld09nfj3y3qxmv8crys4aeut53unfvlqr80nfm already does exactly this? Neat!
apparently what i had in mind already exists: NIP-07