Well how about asking tor users to sign a message from the bitcoin address they registered with instead?
Each time they log in they could be given a unique code and asked to sign a message containing it. That wouldn't cost them anything, and signing a message would be faster than going through endless cloudflare captchas.
registering for a forum account doesn't require a Bitcoin address, only an email address.
Anyone registering through tor has to pay a small bitcoin fee, so all those users have bitcoin addresses associated with their accounts.