Context note: this is coming off of questions Why can't I just offer a bounty for a question right off the bat? (where an earlier version of it was offered as an answer but seemed unlikely to be seen much) and Mandatory delay in offering a bounty (marked as a dupe of the previous question, where I was advised in comments to present it as a feature request)
Proposal: Let immediate bounties happen, but give them a cost of twice what they reward, minimum bounty 100. Possibly crank up the minimum rep necessary to place an immediate bounty (500? 1000?). Those who are willing to wait the two days, you can go under standard bounty rules. This has the following effects.
Immediate bounties are inherently costly. Being able to afford on in the first place means that the user has gained some rep already, and it's not something that can be done lightly or frequently. This should minimize any difficulties with moderation load.
It won't break the rep economy, since swapping immediate bounties inherently takes rep out of the system. Anyone who wants to do it and keep doing it is going to have to generate a fair bit more rep than they're handing out.
It's true, this is nonegalitarian, to a degree. It's essentially another way to take advantage of high rep, added to the long list of such ways (some explicit, some implied). I would argue that this is not inherently a bad thing, and not really at odds (at least inherently) with the way we do things anyway. In particular, I look at this answer to bounty system discriminates against new users .
It will allow a case where a moderately high-rep user who needs an answer to a high-complexity problem immediately can pay for one. It gives a certain class of experts a serious motivation to build rep - because they might want or need an immediate answer (or two, or three) some day.
In general, the sorts of questions that will get this treatment will be difficult questions that require rapid answers and that are not available elsewhere. Giving people a way to "buy" answers to such questions here makes it more likely that they'll show up here, which would seem to be a good thing, overall.