No, not possible. You won't get any interviews.
This really doesn't make sense. You're in Florida, but can't find a PM&R physician to rotate with? I would just find one close to you and do what everyone else does - work extra hours, go in on nights / weekends, get off rotations early and head to the PM&R office.
There are plenty of ways to get a letter, and you have plenty of time to do it.