Your situation is unique since you don't have a pre med committee but for most schools with pre med committees, all the rec letters go to them, and the pre med committee usually has a file of all your rec letters.
I understand you don't have a premed committee but that doesn't mean your previous letter writers should back out on you. Most letter writers keep their rec letters saved that they have written for you. Clearly this PI did in a way reject you, is there anyone else in the lab who can write a letter for you - maybe a post doc or even a PhD candidate, and then maybe ask if the PI can sign off on it, if she wants to of course. If not, then a post doc or PhD candidate is just fine. I don't think medical schools will take it as a bad sign that you don't have a PI rec letter (it may come up in the interview but you can explain it there) but also I mean most medical schools understand that PIs are also busy and accept rec letters from other members of the lab. For my friend, the PI didn't write his rec letter, his post doc did and he got into medical school just fine. Don't worry too much about it. There are always ways to solve the problem and I agree with what everyone said, you don't want someone to write you a rec letter that doesn't want to.
