I am not familiar with your condition but here are a few thoughts:
I would not focus on warming up exclusively one or two fingers at a time because you may end up focusing on them to the point of tensing up
Hanon exercises are good, probably better than scales which would not exercise your 5th finger that much: each exercise focuses on a specific succession/stretch of fingers; quite a few of them work around the 4 and 5 fingers but do typically involve most fingers in each exercise, which again I think is a good thing given what you have. Choose some exercises that are easy for you to play so that you do not put any tension in your wrist, hand, and finger as you play. Play them slowly enough so that you can focus your attention to playing them with ease and not tensing up: this isn't a racing exercise; just exactly what you are asking for: a gentle warm up. Simply increase the speed if you can still play them without any tension.
You may want to play some successions of octaves with 1-5 or 1-4: these will bring you a different kind of movement than Hanon or a regular scale, requiring a gentle stretch of the hand, which I think may help with your situation.
Whatever you do past the warm up, you should actively fight any build up of tension, especially tension in the arm, which will propagate to your fingers. Taking breaks every 10-15 minutes or so is probably wise.
After playing, consider shaking my hands as if you are drying them.
If you still play tennis, work on your lifts and spins: having a rotating movement of the arm as you hit the ball will decrease the build up of tension in your forearm. For climbing, short of using your legs more, not sure what you can do!