Should work fine.
You can get studded tires in BMX size: the Schwalbe Marathons are available in 406, fr'instance. Way more spendy than DIY, though.
If you'd like to try before you buy, I've got a set of 406x42 Winters in my basement. I'm not interested in selling 'em, but you're welcome to borrow them for a week and see if you like them.