Re: vinyl flooring replacement
"Factory built house", do you mean manufactured home (aka, mobile home ). If this is the case and the floors are original, then it is not likely that there is any floating floor system. Trailers are all glued and stapled together, it's the only way they'll stay in one place while traveling down the interstate to their final resting place. Given this scenario, it would be perfectly acceptable to float the existing floor with leveling compound and overlay with the new flooring.
The only "floating floor" that I'm aware of in a sheet good was something called "perimeter glue" vinyl. A 4" to 6" strip of adhesive or tape was placed around the perimeter of the room and the vinyl stuck down to that, there was no glue/tape in the center of the field as is the customary installation method. Perimeter glue installs are no longer done. If this is your case, then the old vinyl can be removed relatively easily and the new laid in it's place. The only time that underlayment would be necessary is if the existing subfloor/underlayment were in poor shape. You can test to see if this is the case by poking the tip of a knife at a sharp angle into the surface of the vinyl and lifting. If you can get it to lift it's perimeter glue, if not then it's likely fully glued down. Only do this test if you're certain you will be replacing the floor.
Without seeing the floor for myself, I can't offer an opinion on the two bids you've received.
I suffer from CDO ... Its like OCD, but in alphabetical order, LIKE IT SHOULD BE!!!