Should I just let the bounty run out and award it to the best existing answer? Can I request an extension? Are there other options?
See How does the bounty system work? for more information. To answer your specifics:
- Bounties cannot be extended.
- You can award the bounty to any answer that you didn't write yourself.
- If you don't award the bounty, and there is an answer written after the bounty started that has a score of 2 or more, that answer is auto-awarded half the bounty.
- If there is more than one such answer, then the answer with the highest score is auto-awarded, with ties broken by earliest posting date.