Dear 3 Disney Bugs
The first thing you should check on is moisture. Is your basement floor damp in the winter? If your home is relatively new, 20 years. Chances are there is a vapor barrier under you concrete slab. It usually just a sheet of 6 mill plastic under 1 inch of sand and over 4 inches of gravel. Here in California it is a code requirement. Even if you have no vapor barrier if you have no moisture on your slab in the winter or Rainey season you shouldnt have a problem. This is one way to test for moisture. Sit something on the floor for a week or two in the rainy season, like a 5-gallon bucket or a cardboard box. If you pick it up and it is damp you have a problem with moisture. With all that said and not knowing how uneven your floor actually is it sounds like the guy that quoted you
$ 5,000 has nothing more in mind than whats in your wallet.
The tolerance that I go by is a hump in the floor cannot be more than ¼ inch in 10 feet if you can reach that tolerance you shouldnt have a problem. What you will need is a straight edge 10 feet long sit in the middle of the hump and allow no more than ¼ on each side of the straight edge. Next take your time and go though your whole floor and circle each hump with a lumber crayon. Mark on the concrete the amount of concrete that needs to be removed. Low spots are not a problem the can be filled by the tile setter. And you should not be charged unless were talking ½ inch or more. Next you will need to find someone in your area who does concrete grinding. Make sure there grinder is equipped with a vacuum or your home will be filled with concrete dust. You can check with Redi mix plants, Yellow pages and with concrete subcontractors that specialize in production Homes. The usually have a crew in house that does concrete grinding.
After they have finished grinding take your straight edge and go over the spots the have grinded. Dont let them get away before you double check their work. Try to get a total price for grinding the whole floor not an hourly price.
A couple of things
The Larger the tile the harder it is to lay and that means dollars.
A good tile setter will have less lippage than an inexperience tile setter.
I have no tolerance for lippage.
Good luck
Searl