My $.02 says to rip out whatever was put in, and when the new floor is done, make sure you can get some guarantee that what was done initially doesn't happen again. This is assuming 1) you can be told what was done wrong the first time, and 2) whatever went wrong the first time won't happen the 2nd time.
I'd stick with Armstrong if you like it as long as they're paying to fix the problem. Since it's under warranty, I don't think you could easily get your $$$ back. As a judge (which I'm not), I'd have to give the mfgr. the opportunity to fix the problem at LEAST once, then, if it doesn't work the 2nd time, THEN, I'd say you have a good case to get your $$$ back (that's 2'xs that Armstrong would have messed up). Also, I'd make sure your warranty begins with the installation of the NEW floor and doensn't keep going based on the 'old' one. It is, in fact, a completely re-done floor. Insist on it! If the 'old' underlayment is coming through, it should be removed. (Well, is it the underlayment / luaun, or are you referring to the backing on the flooring itself? They are 2 different things.) If the 'backing' on the vinyl is bad, yes, get something else or get some guarantee that they'll give you a new floor (and a new warranty) if it happens again. If possible, for peace of mind, see if you can get a different floor. Maybe Armstrong is saying their vinyl-backing is OK but the adhesive was bad, or the wrong kind. I've never heard of a problem with the backing. Do you know what it is???!!!
And lastly, if it was my floor, again, I'd have the old one removed.
QUESTIONS: What is the underlayment that you had installed with the 'bad' flooring? What is UNDER the underlayment that you had installed with the 'bad' flooring?