I assume that the check for answering protected questions is simply implemented as something like this pseudo-Python code:
threshold = 10
threshold += 100
if user.rep < threshold:
raise NotEnoughRepError("Sorry, you need more rep to answer protected questions.")
By receiving the association bonus, and then lowering your rep below 110 with your bounty, you caused this check to fail.
Is this a bug? Technically, this behavior does contradict the text of the protection notice, which says (emphasis mine):
"This question is protected to prevent "thanks!", "me too!", or spam answers by new users. To answer it, you must have earned at least 10 reputation on this site."
Strictly speaking, it doesn't say you have to have at least 10 rep at the moment, just that you need to have earned that much rep on the site at some point in the past. So, to exactly match this literal wording, bounties, downvotes and any other mechanisms of rep loss should also be discounted.
Of course, this is almost certainly not really what was intended. Indeed, the FAQ post "What is a “protected” question?" is a bit more specific:
"Users with 10 or more reputation can answer a protected question. However, the +100 account association bonus is ignored for this check, so you must have earned 10 or more reputation on that specific site to answer a protected question."
This description does match the way the implementation seems to work: you need to have at least 10 rep on the site, excluding the association bonus, if any. As a corollary, this means that you need to have earned at least that much rep on the site (and possibly more, if you've also lost some).
(Actually, if the threshold is indeed 10+ rep, then it seems like you'd only need to earn 9 rep to be allowed to answer protected questions, since rep starts at 1. That should be doable with one question upvote and two accepts / approved edits. I'm not sure if anyone's actually tested this, though.)
I suppose, if this really mattered enough, the protection notice could be changed to say something like:
"This question is protected to prevent "thanks!", "me too!", or spam answers by new users. To answer it, you must have at least 10 reputation (not counting the association bonus) on this site."
However, this doesn't really seem worthwhile just to fix a minor discrepancy in an obscure corner case, especially as it would probably make the message a lot less clear to many new users, who may not immediately recognize the term "association bonus".
Anyway, on a more practical level, I see basically two obvious ways to avoid this problem:
If you don't want to lose privileges when you give away most of your rep as a bounty, then, as the old doctor joke goes, don't do that. Losing privileges if you give away too much of your rep as bounties is a deliberate feature of the bounty system; bounties are meant to be rare and significant, which they wouldn't be if they didn't have a cost.
If you've already started the bounty and lost the privileges, the only thing you can do to regain them is to earn more rep. Fortunately, the bounty system doesn't allow you to go into negative rep, and earning (up to) 110 rep isn't really that difficult — a few decent questions or answers will do it easily.