I've used no-pest strips pretty effectively over the years, however I've also known people who have killed animals over using them. I think your idea of putting one in a secure tub in the garage is a pretty good idea but you might not want to use the whole strip. Before the invention of PAM and other mite treatments, people would cut a 1 sq inch chunk off of a no pest strip and put it inside of a film canister that had been drilled with a lot of small holes and use these to treat for mites. That might work for you but remember that poisons will not kill mite eggs, as far as I know only heat will kill the eggs.
Also, snake mites are species specific and will only infest snakes. If you got snake mites from your bedding, they only way they could have gotten there was if the bedding was stored near an infested snake and the female mites were seeking a place to lay their eggs. This can happen if you buy your bedding from a pet store. It's much less likely to happen if you get it from a garden supply store.
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