The LOR is a snapshot of who you were at the time the letter was written, not a binding contract. The "transaction" has already been made at the time the letter was written as a result of your previous actions, so you don't owe them anything *more*. So don't feel guilty for leaving.
I guess my primary concern would be: If you don't get accepted this year, will you feel comfortable asking this person for an updated letter next year? Have you continued to produce the same high-quality work on the job in the few months after receiving the LOR?
I've been an even more sticky situation before — I announced I was leaving, *then* I requested an LOR for a job app, *then* I left the job a few months later. And later that year I asked for *a second* LOR for my med school app. Obviously this whole series of events was quite delicate to execute and could have been planned out better. But I think it was received well because I left on good terms with everyone, followed through on several projects in my last few months on the job, and was respectful. I agree you don't want to leave *immediately* (say, within 1 month) after getting the LOR, just for professionalism's sake. But as long as you continue to produce the same high-quality work, my gut feeling says you should be fine leaving.
(Whether or not this is a *good* LOR is another question, since you mention working with many docs who may or may not see you every month. Have you considered asking your manager for a LOR?)