There are a thousand ways to remove stripped screws, though it varies with conditions.
Step 1 is can you get a flat head screwdriver or some other screwdriver (bigger size or smaller size from the one you used) down in there to sort it out? Do remember a few degrees tighter before reversing direction can also break a sticky joint if that is what is happening. A bit of light pressure on the plastic case (either squeezing from above to remove pressure or in the middle to add it) can also help matters.
Some also like to melt a bit of plastic (pen lids tending to be the thing that is sacrificed and see if that holds for it. Some glue might also do something for you but be careful with glue (superglue and polycarbonate, which was at least in the Switch shell, is not a winning combo for you here)
They are probably going to be too small for the things like stripped screw removers, or at least too small for anything you will find on the easy market that tooling won't cost more than a new joycon.
At this point you get to drill things out. If you can then use a drill press rather than a hand drill.
If you don't have such toys (I also imagine you don't have anything like an Archimedes drill) and don't have a neighbour with one (they are among the most basic machine tools that woodworkers, metalworker and plenty of others will have) then with a hand drill see if you can take some of the guesswork out by lining things up and clamping it to the table such that you just have to feed it in with it guided by whatever you have at hand -- if you are asking this sort of thing you are probably not one of the forearms of steel and can hold a drill steady for hours at a time types I sometimes meet on job sites. It will take a while to set up and be quite tedious but better that than a new joycon or having to repair things after you mess them up.
The idea will be either
1) You drill a line of holes in there such that it now fits a flathead.
2) That you drill far enough down with a big enough drill that the head of the screw is no longer attached. At this point the screw is just a rod and you can remove it, and unscrew the remaining shank with a pair of pliers or something.
Depending upon how much of a mess you have made of the screw head you might also have to use a punch to put a divot in there to guide the drill to the proper location rather than have it wander off.
"I don't have a bit small enough"
...
Buy one if you can but that might be hard at this point in time. Drills are kind of special but also not -- if you file (or rub on concrete for long enough) a flat section in a old nail it will drill, however I try not to do that for something this small and delicate.
If you can spin your drill leftwards during this (if not doing the above then do it with a left hand drill, something you are not likely to have -- I don't have any right now and I have most drills in several different types) it might even catch and spin the thing out for you.
If you can expose the shank of the screw and get a hacksaw blade in there you can cut the shank and thus remove the top of the screw. Some might twist a case to get here, and if you managed a few turns before it stripped then you might have space.
Being plastic you don't have weld options (the melted plastic screwdriver approach being the one here) and I will skip chemical stuff (plastic will not respond to chemicals that will eat away the screws quite happily, however said chemicals will probably also mess up the PCB and also cost a reasonable amount)