Is it possible for a bounty-awardée to reject a bounty in order to prevent the decrease of the bounty-starter's "reputation"?
No. Bounties are there to give questions additional exposure and are removed from the 'donors' reputation at the point they are set up. Even if the question involved does not gain any answers as a result, the bounty is not returned to the donor. The question will have been 'promoted' by the bounty (there is a dedicated bounty list which is much smaller than the full list of questions, or even of unanswered questions).