@jonny I don't know providers, but unsubscribe is the RFC 2369 List-Unsubscribe header. If you can get anything capable of high email volumes, where you can set headers, you should be able to do this semi-manually to bide time until you can get a good system in place.
So, having someone vaguely-technical (really, they just need impeccable BCC hygiene and the ability to set headers when forwarding emails) be the designated Mailing List Operator might suffice.