Personally I'd just get straight back to climbing using front three only for a bit. Pretty sure there's some science saying training the other side helps recovery i.e. injured left leg, training right leg improves recovery. Not sure if the same applies to neighbouring fingers.After a week or two I'd try some more open handed slopey problems (one the easier end) with all four fingers and ease the pinky back to full use.
perhaps a better explanation of "strain would go a ways. Pulley, Joint, ligament, muscle? All are real options, and slightly different treatment
Climbing on jugs is often excellent rehab.
If it's around the outside of the finger, then I'd be surprised if its the pulley. Your description sounds like it might be a collateral ligament issue or a c2/c3 issue. Climbing on jugs won't really hurt it if its the ligament, but it won't help it much either. https://theclimbingdoctor.com/portfolio-items/collateral-ligament-sprain/
If it's around the outside of the finger, then I'd be surprised if its the pulley. Your description sounds like it might be a collateral ligament issue or a c2/c3 issue. https://theclimbingdoctor.com/portfolio-items/collateral-ligament-sprain/