It's a pulley, it's always a pulley. Always.
Ok, firstly there are no quick fixes. Pulleys have poor to no blood supply they do not heal like a muscle tear. You need to be patient. That said you appear to be fortunate in that the symptoms as you have described them are at the mild end of the spectrum despite the fact it has clearly been bothering you for a few weeks. Essentially at this point in your recovery your priorities are not making it worse and not stopping entirely. So keep climbing, tape your PIP joint so you can't bend your finger, do not tape around the pulley itself. Climb carefully and in control and open handed. You don't want shock load it. Don't bone really hard on anything. Select your problems carefully and immediately sack off anything that hurts it. If things seem to be improve think about adding in some controlled static open handed dead hangs in a couple of weeks or so.
I don't think this adds much to the advice already given (did someone mention it was probably a pulley?), but having it in both middle fingers and open handing everything sounds similar to issues I've had. Is your middle finger more flexed than your index and ring when you drag a 3 finger edge? My middle finger tends to roll outwards towards my ring finger in that position and I think that torque is what fucks them up. It might be worth trying to half crimp things as well to give it some lateral support from your other fingers.
