Just some ideas for thought...
I like the idea of the overflow rep from the rep cap just going into a users personal bounty pot.
Limits for overflow bounty earned
The amount being overflowed into the rep cap pot should have daily and total limits.
Say maximum of 50 bounty rep per day into the pot, and a total maximum of 1,000 bounty rep allowed in the pot at anyone time.
Can't earn your own overflow bounty
As this is overflow rep a user wouldn't have had added to their actual rep, I think you'd have to disallow a user being able to earn the bounty they placed from the rep overflow pot.
Users could just put the bounty on a question they are good at/have an answer prepared for and whack a good answer out, and earn rep they were originally capped from earning.
High rep users only
I also think that the overflow bounty pot should only be allowed to higher rep users. Say 5k (for figures sake). It would be nice to give it to everyone, but I think it will be enough giving it to those with a good chunk of rep already.
Users with > 5k are obviously active having such total rep, and so know the site etc and more likely to place it on decent questions, and less likely to abuse it somehow.
Lower rep users still gain from this too as they can win the bounties passed on by higher rep users.
Cap on overflow bounty earned per day
There should also be a maximum limit you can earn from bounties placed from overflow rep bounties.
Current bounty you can earn as much as you like, which is fine. As it comes from your own site rep not many people throw bounties around, and also the site's total rep is just shifted from one user to another, and not additional injected creating an imbalance in the total high and low rep users.
However, with the overflow bounty, the overflow rep that was originally lost is now injected into the site.
So users earning bounty from overflow rep should only be able to earn a total of (say) 500 per day, as with this bounty there is some potential of creating a lot of high rep users.
Though from this new feature I really don't see a sudden burst of hundreds of bounties every day and every Tom, Dick and Sally becoming high rep users.
It still has to be earned to be passed around and won.
Limits for placing overflow bounty
I think users with overflow bounty rep should be allowed to place up to 500 per question, or less and split the total overflow bounty they have (1,000 limit suggested) over as many different questions however the users wishes.
I also don't think many users would throw their overflow bounty points rashly on any old questions. Yes some will, of course, but do you think more or less than currently high rep users throwing their real rep around?
While overflow rep bounty isn't from their site rep, it has still been earned, and so I think people who don't throw rep around now will value it similar to their site rep. Those who would throw it around, well, does it matter so much as long as it's not "gamed" and is earned?
I don't see much of a difference. People with > 5k easily have 150 real rep to throw around a few times as it is now, so I don't see them changing their attitude to their rep just because they got a bit extra in a pot for bounties only.
Likely a big project, but benefits Stack
All that said, it is a large ish project to do all this.
With the new requirements and all sorts of functions, checks and monitoring in place, even the basics like when you place bounty people with the overflow pot you now have to be presented with some kind of choice of "from your site rep" or "bounty pot" and that having to calculate what they can place based on the total of each individually.
I think it would be quite a lot of work, but certainly offers benefits..
This additional overflow rep is earned, by those who have the overflow bounty pot, and those winning the bounty. As this is earned by providing good questions and answers, it surely only benefits Stack and it's users/visitors.