Have them try to nail it first. Rare is the occasion you cant nail concrete strip through two layers of tile. Dont glue the tackless. It will be a futile effort because the pressure of the stretch will most likely pull up the tile the tack strips are glued to. Pre-padded carpet is certainly an alternative like Alice suggests. Other options are shooting in a layer of plywood over the tile that you can nail tackless to and install carpet on if nothing else works. We have successfully drilled through asbestos tile with slow rotary masonry drills/bits. This did not create dust from the tile because of its resiliency. It just flaked off chips that were easily shop vacced up. I do not recommend a high speed drill though. If the tile has resiliency then it may work for you, but if brittle, then may cause dust because of its hardness and is not recommended.

I would put down cement board, then install the tile over that--but be sure you can find a transition strip that will allow you that much of an elevation difference between rooms-since it sounds like there are several layers already.

Depending on the City Code, a homeowner needs a permit to demolish or remove anything from a home built before a certain period. The burden of proof is on the homeowner to prove that there is no asbestos. I recently had a leak in a wall from an upstairs neighbor and before a fairly small portion of drywall could be removed, I had to get it checked by a licensed testing agency bore anyone could remove it. . It had asbestos so the licensed hazmat crew had to come out - total cost to remove was about $2000 including getting the test. Tests and removal cannot be done by the same company for obvious reasons in terms of potential conflict of interest. Same thing when my popcorn ceiling was removed - test was required and a hazmat squad had to remove it AND I then had to have it tested again for asbestos. When I was looking for flooring several years ago, the flooring store asked me when the original vinyl tiles had been installed and told me that I would need to have them tested before they could be removed. The trash hauler was probably required to ticket any construction materials being disposed of with the normal trash - especially in that era. Having asbestos removed properly adds significantly to the expense of just tearing it out. Everything has to be taped with negative pressure - the debris has to be properly wrapped up and then disposal is done at specific hazmat sites which are obviously more expensive than hauling out to the regular trash.

Carpet is the WORST product for allergies, asthma, COPD, etc. There is nothing wrong with asbestos tiles. They are stable. They do not 'give off' asbestos. It is only when they are broken (pulverized) that things become a concern. it sounds like you have no intent to pulverize these tiles into beach sand, so you are more than welcome to leave them exposed.