My error, I should have said "tile guy" instead of "tile store".
A lot of the tile stores do the same thing as HD/Lowes, just hire out to a cheap guy, although in my experience the carpet/tile stores do tend to stick with the same guys that do a consistently decent job. HD/Lowes actually works through a referral service in a lot of areas where they hire whoever is available.
You might look for a contractor who does "handyman" jobs to see what they'd charge just for the tear-out, and see if they do any tile work. Just make sure you see their work or get a reference from someone you know.
Contractors that just do tile work and arent related to a store can often do better quality work and the total price might be reasonable. Plus they'll know who has good tile selections and you can shop around a little with them. Some handyman guys will do a pretty good job with a simple floor or shower retile. If I was doing something complicated or with expensive products I'd go with a regular tile pro.
Only other thing I can suggest is that if anything with parts is involved...doors, tubs, fireplaces, etc...that you pick up all the small parts and collect them in a bag for reinstallation as they're removed. I've had an awful lot of ridiculous little cheap parts that were impossible to find get rolled up in bags of construction junk and thrown out. When we redid our bathroom, I ended up missing two tiny little metal tabs that hold the doors in and a mystery gasket that sealed the tub to the jet pump. I actually still havent found any source to the fireplace door parts and it took me a week of calling people and trying different gaskets before I found a spa guy who knew the motor and what the gasket looked like and found me one.