The amount of rep spent on bounties should not affect a user's ability to answer a question, since bounties are themselves a sign of useful participation.
If anything, a history of placing bounties should have no bearing (or even a slightly positive bearing) on whether a user ought to be able to answer a question.
Solution
Allow the >=10 rep check to be performed on gross rep, not net rep. Keep a column of gross rep in the database, and only change it when a user places a bounty (so gross rep will always be a users rep + any rep spent on bounties)
Example
User with an association bonus, gets 10 upvotes on an answer for a total of 201 rep, then places a 200 rep bounty. Their rep is 1, but their gross rep is 101, well above the 10 rep threshold for answering a highly active question.
Is it worth it?
The only remaining question is the effort worth it to cater to a relative small number of potential answerers. That's a question I can't answer.