Speaking about this as a "grace period" doesn't make sense to me. I think Jeff's answer is quite reasonable.
On the other hand, if there were a "Judging Period", that might make everyone happy. What if the question were automatically Locked or Closed for 24 hours at the end of the bounty period -- no more answers (no more votes in the case of Locking), nothing except for bounty awarding, until a day had passed? It would then be re-opened and be a normal question in all respects.
The major problem* is apparently the fear that the bounty will be awarded before the best possible answer comes in. Having a period where the bounty can be awarded but no further answers can be added would give the asker plenty of time to read and evaluate the answers without that worry. The bounty period would continue to be set and stable.
It seems to me that bounties set up a kind of contest, and it's normal for a contest to have a hard due date for entries, and then take some time after that to judge and choose a winner. I think it's a good attitude on the asker's part to want to judge carefully.
This might make "the bounty logic even more complicated" though, I don't know.
UPDATE: Thinking about this more, I now agree with Anna and Tobias that closing the question isn't a good idea. I was putting too much weight on the "contest" aspect of the bounty. The only way I see around this would be to still accept answers during the hypothetical Judging Period, but to not allow those to be awarded the bounty. That would certainly make the system more complicated and confusing, and so is probably a non-starter.
*Besides "I forgot", which isn't really valid.