Having had injuries that had to be rested, then nursed and then strengthened before starting to return to bowling, I'd wonder about the inflammation in/on your wrist. UNLESS you have had both X-rays and an MRI, I would worry a whole lot about the potential damage done, before I'd begin any regimen involving exercise of that wrist and bowling.
There are far too many pieces of bone, muscle, tendon, ligaments and God only knows what else in a wrist, including the connections back up to the forearm and down into the hand.
I'd strongly urge some kind of confirmation of the source of that swelling/inflammation before doing anything else.
Normally, you rest the injury until it feels good doing everyday things; then you strengthen it slowly over time. (The younger you are, the quicker you can usually do this; the older, the more gradual the build-up needs to be.). Then you test it, EMULATING the more physically stressful motions of motions, like bowling. Then, and only then, do you try the ultimate, actual bowling. Hand, wrist and forearm strength AND flexibility are some of the keystones for holding heavy objects, like 15 or 16 lb. bowling balls.
As someone in the midst of the final stages of recovery and return to bowling, I can attest to every word I've written. Being older, I tend to the cautious side of the recovery process. Realizing that waiting another 2 or 3 WEEKS may allow you another 30 - 40 YEARS of bowling may allow you to do exactly that.
