https://wicg.github.io/local-network-access/#integration-with-permissions defines 2 permissions, but it's not super clear to me when the user is prompted and what happens as a result of the users choice.
I see that 5.4.4.3 of https://wicg.github.io/local-network-access/#local-network-access-check does have some "prompt the user" language, so presumably that corresponds to "local-network". But I'm not sure where "loopback-network" is handled.
https://w3c.github.io/permissions/#prompt-the-user-to-choose gives you something to hook into, and maybe the couple of examples at https://dontcallmedom.github.io/webdex/p.html#prompt%20the%20user%20to%20choose%40%40permissions%25%25dfn are helpful.