Hi---my parents recently bought a lake house and were faced with the same issue---they opted to remove the asbestos. They had wood-looking porcelain installed throughout. It's the next best thing to natural stone---no need to worry about ever having to replace it due to the elements... Good Luck!
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I would try and talk to one of the sales reps for the company that makes the flooring...

Thanks Natalie, we're trying to do a lot of the work ourselves and am not sure we can remove the tile or dispose of it if we do!

Love the wood look porcelain...

well, I don't know if this is too late but we put Allure Traffic Master floor from Home Depot in our basement in September. There are not much temperature extremes here in Seattle. We had a halfway chipped up floor w/possible asbestos situation.

They (my husband and boyfriend in law) floated the floor with thin concrete to make it level and put the floor over that with no removal of old flooring. The only thing is, the directions say to use a floor pressor thing to really seal the seams and my husband doesn't think it's necessary. I am a direction follower and it makes me nervous.

Anyway. They did get a little of the sticky gluey stuff on top of a few tiles and I could not remove it even with a razor blade. I used a little baby oil and it zipped it right off. It looks so nice! Too soon to say how it will hold up, but there are no signs of anything bad so far.

I have also used the porcelain wood look tile (with the thinnest line with the most synthetic grout) in a house in Louisiana and it was quite beautiful and easy to care for.