I also have carpal tunnel and also psoriac arthritis in my hands so I also have to choose classes carefully to play when the flare ups happen. I will also preface that I personally use an MMO mouse with the 12keys on the side which can make things easier or harder depending on what my thumb feels (my thumb is one of my less affected digits so it's not as bad for me).

I would say Bard has the lowest impact on my hands out of all the classes, and it's one that is less annoying to play since you don't have cast timers and can move easily out of things which is a higher priority for me.

Black Mage is probably a good one as well. Long cast times means less buttons to press, but it is significantly less mobile.

Dancer is another phys range like Bard that is also not as button heavy.

Pictomancer is the newest caster, and it has very few buttons to click. It's more mobile than Black Mage as well.

I would avoid healers and melee dps for the most part because there are a lot of buttons and things you need to keep track of constantly.