This was always my problem with bounties on Stack Exchange, so that’s not quite what I had in mind. I was thinking you’d consume “bounty points”, not “reputation”, i.e. something which exists exclusively to be awarded as bounties and serves no other purpose. Of course users can make something serve some purpose we didn’t intend, which is why I propose keeping the number hidden.
Even if we were to use moderator reputation (which, again, doesn’t make you a topic expert, so may not be the best idea), I would still propose separating what’s used for bounties from what your reputation is. So, for example, 10 good edits might give you 1 mod reputation and 1 bounty point. Spending the bounty point doesn’t affect your mod reputation.
Although it feels like overkill to do all of this just for bounties, so perhaps it’s just something to keep in the back of our minds to potentially combine with other functionality.