you could build a false wall inside the closet
it doesn't sound like you want do a lot of carpentry but all you have to do is buy a sheet of dry wall cut to the dimensions you want and make something simple to support it
it would solve your problem of cutting into the door for venting and when you sell your house you just throw the piece of dry wall away and no damage is left also if your kids don't know the closet dimensions just throw some clothes or boxes in front of it
Also if you have an attic you can vent into there it will still create a hole unless you have a light fixture in there if you do just take the fixture out and you have a hole to the attic
it would depend how your closet is set up but it would be relatively easy and might solve a few of your problems.