Virtually all of the martial arts use the hands in some way. Even Taekwondo, which uses mostly kicks during sparring, will use the hands to block and punch. Whereas, grappling arts use the hands to grab onto the gi or wrists or whatever. It's not uncommon in Brazilian Jiujitsu or Judo to sprain your pinky and ring fingers due to the fact that your grip entangles the fingers in the gi, and people are using a lot of force to escape that. Weapons based martial arts like escrima and silat use sticks and knives during training, and often there are impacts on the hand and fingers, causing sometimes significant injuries.
You might be thinking about martial arts that don't do much impact at all or even physical contact in general. Something like Tai Chi or Capoeira comes to mind. But in Tai Chi, it's not just solo forms practice you're doing. A "good" Tai Chi program will have push-hands practice, punching bag exercises, chin-na (grappling), throws, and even free sparring later on. Whereas, Capoeira doesn't make hard contact (usually) during practice, but you're often using your hands to push off of the ground, which puts significant load on the bones and tendons in your hand.
So one recommendation is to just stick with solo, forms-based martial arts for a while, and excuse yourself from any contact-based exercises within those martial arts.
Tai Chi is a good candidate, but really any martial art that uses forms is good. You just have to explain to your instructor that your hand is damaged, so you can't do any contact based exercises with it. Tai Chi is perfect, because it's very common to see classes structured whereby you have a forms class that's separate from push-hands and chin-na classes. At least in many Tai Chi schools. And Tai Chi is one of those arts that requires long term, almost obsessive focus on your form, so it's not wasted time in other words.
There's something else to consider also. That is, you can get a simple arm sling and make sure your arm is strapped to your chest the entire time. With your arm in the sling, it's immobile. You can't use it at all during class. So you'll be forced to work one-armed the entire time. And that may be useful in different ways. It would allow you to continue doing whatever martial art you want, so long as it can be done with one arm, and so long as you're careful.
Here's an example arm sling product, although I don't really know anything about its quality (Amazon reviews rate it highly, and it's cheap)...
Your hand may still be impacted, though. Like someone can punch or kick it, or you can fall on it. So it may not be ideal. You can wrap your hand in tape or wear a punching pad in addition to the sling, though. That would increase your protection. But it's not fool proof. There is still risk, and that's something you have to weigh.
By the way, another way to make your arm immobile would be to tuck it inside of your gi belt worn around your waist. So instead of being strapped against your chest, it's strapped at your side. That may be safer for you, or not. It depends on what martial art you're doing. You also want to make sure you have a wrist strap (with tape or velcro or something) that wraps around the gi belt to prevent you from withdrawing your arm from your belt.
My thoughts anyway, for what it's worth. If it were me, I'd lay off of it completely for 3 months like the doctor says, and then reevaluate then. I would definitely get the arm sling in the meanwhile, to prevent any accidental use. I'd do basic exercises like walking and running, crunches, one-armed push-ups (using knees or something), one-armed pull-ups (actually, pull-downs), and so on. That way, my strength and cardio wouldn't deteriorate too much. The doctor may recommend rehab, and I'd do that afterward. For martial arts practice, I'd stick with forms and solo drills. I'd probably take a hiatus from formal training in a school for those 3-6 months. Long-term, I'm pretty sure rehab and strengthening would allow me to return to full martial arts practice, but with some extra safety precautions, being mindful not to re-damage the hand. That's just me, though.