Are you sure its a pulley injury (as apposed to flexor unit)? From my limited experience it sounds more like a tendon rather than pulley problem... getting it checked might help find out?
As for your actual question of
if it would be suitable to train on the fingerboard but sticking to open handed only?
I would say yes, seeing as there is no pain!
My recent flexor unit strain cleared up whilst I continued to climb. I made sure I used my pinkie and was conscious about grip/hold types. Buddy taping helped remind me to do this, but as has been said elsewhere should not be used for much more than reminder. However, I was set back twice by getting over excited and tweaking it with and then without the tape so just take it slow!