Carpet over tile without damaging tile..
You COULD use contact cement to glue down tackstrips around the perimeter of the room, but I don't know how well that contact cement would hold up under carpet tension. If the installer knew that the tackstrips were held down with contact cement, he wouldn't use a power stretcher to install the carpet. Maybe tell him to install the carpet just with a knee kicker. They're supposed to use a power stretcher for a proper installation, but thousands of carpets are installed with just a knee kicker every year without any problems. If you do use contact cement, you'll need to use toluene or lacquer thinner to remove the contact cement from the tile later on.
I've installed carpet over concrete with a knee kicker where the tack strip was glued down with LePage's PL Premium construction adhesive, and that was fine, but contact cement is nowhere near as strong an adhesive as PL Premium. I think, tho, that contact cement would probably be strong enough as long as you don't stretch the carpet much.
Another option, which I would probably do myself is to install the foam back carpet, and just use double sided carpet tape to hold it down around the perimeter of the room. You can use double sided carpet tape to stick down any carpet to tile, actually.
Alternatively, use plenty of double sided carpet tape to stick your underpad down to your tiling, and then full spread glue your carpet down to the underpad.
You'll be able to pull up the carpet tape cleanly. I've pulled it up off of vinyl composition tiles. It's strong tape, but it does come up cleanly. They should sell it anywhere they sell flooring installation supplies, and your local hardware stores or home centers should have it.