Yes, NIP-07 signers don't allow signing of arbitrary message at the moment, so a PK is unavoidable:

https://github.com/nostr-protocol/nips/issues/292#issuecomment-2727709548

NCR is browser only for PK ops - nothing sent to our server.

Or you can wait for Cashu.me to support it (also local use of PK).

Or you can use 0xchat (also uses PK for login)

Or leave it and those 10 sats are burnt forever.