Given that the point of this type of bounty is explicitly to reward content that already exists, the primary purpose of the waiting period - to give people a chance to respond to the bounty by providing new content - is irrelevant.
However, the secondary purpose of the waiting period, to draw attention to interesting content, would be undermined by this change, which is a shame, since this bounty identifies great content that's deserving of attention. Therefore, instead of a waiting period, there should be a period after the bounty is awarded (a week, I guess) during which "Reward existing answer"-bountied-questions remain on the "featured" list.
Regarding the tertiary purpose of the waiting period, to prevent abuse such as rep transfer for its own sake, that could be addressed by treating the bounty as an "open" one while it's still on the "featured" list for the purpose of counting how many simultaneous open bounties a user has, thus limiting the velocity of such transfers. In fact, making this period a fixed minimum of more than a day for this type of bounty would actually make it even harder for users to use it serially to dump reputation on another user.
In addition, it would make sense to split the featured list into two, representing the two types of bounties: one for bounties that are open to solicit new answers, which would be associated with the "unanswered" or "needs answers" part of the site, and one for bounties that have been awarded to reward an existing answer, which would be associated with the "recommended" or "hot" part of the site.