Any recommendations for relaxing/training that muscle?
This is certainly not a cure, but it should help. Spend some of your practice sounding bad. Let me explain:
Background Analogy: There is a trick where someone holds your arms as you try to raise them for a minute (not a guitar thing). After the minute is up, the subject's arms tend to raise involentarily for a brief time period.
Sounding bad: Guitarists tend to put more into their "left hand" (fingering hand for those who play with the other hand) then they need too. If you practice not pressing down with your left hand for a fair protion of your practice so that you get a thunky buzzy sound (but not all of the practice, of course. One needs to hear one's guitar playing during atleast some of the practice), you will be developing a lighter touch and learning just how little it can take to sound good on a good guitar.
Additional advice: It also sounds to me like allot of your strength is comming from your thumb. Use your arm to support or replace that work your thumb is doing. You can press the strings down by pulling back with your arm. For a parlor trick, I show my students without using my thumb at all, but you should use your thumb some. Your thumbs role should be moved to a more supervisor capacity where it feels the guitar and helps tell your fingers what to do.
Human Advice: If you can avoid surgery, and or cortisone shots, I say avoid it. They are last resorts. The surgery for metacarpal injuries is particualry grousome, and you can never go back if it makes it worse. That said, I am sure there are examples out there where the surgen worked wonders, and you should at least get an diagnosis if the pain persists. Cortisone shots are often provide only temprary relief, and it is possible that you are masking the pain instead of eliminating it. Then it is possible to make things worse instead of better when you continue to play without pain. Also Do not continue to play during the times when you are in pain. This seems obvious but I have seen many including myself who are tempted to continue to play anyway but it is dangerous behavior.
All this pain is why your good, thoughtful, grumpy, first guitar teacher may have continually reminded you not to wrap your thumb around the neck. (What, it was just my teacher? I think he would have wacked my thumb with a ruler, if that sort of thing wasn't passé then. Pain to prevent pain.)
There is other technique advice to be given for the piano, but I leave that to a piano player more experienced than I to give. I think it involves keeping your wrist raised slightly and a fluid relaxed striking movement.