Andrew,
Hey, sorry to hear you're injured. Been there a few times in the last 25 years of climbing. I'm sure you're hearing lots of different advice so thought I'd lend mine.
You are doing the right things now to heal up. Down the road
patience will be the most important part of your recovery (and the most difficult). I've healed in 3-4 months. YMMV. Tape the joint for support, and as a reminder to take it easier. Slab climbs will help take the intensity off your fingers. How about hand cracks too? One of the Climbing training books has a section about healing those pulley tendons. They recommend climbing lots of big open grip holds for weeks.
Lastly, I have come to believe that gym climbing is more likely to cause me injuries than climbing outside. I don't climb in the gym anymore and haven't had any pulley injuries in a decade. Something about all the thin crimps without the natural variety of position one encounters outside. For what it worth . . . Good luck, Tom