TLDR: Go with SmartStrand first, nylon second, polyester third.

I sell flooring for a living. I have SmartStrand in my house, with a potty training puppy, and it has performed beautifully. I personally would look to do SmartStrand first, but you wouldn't go wrong with a Shaw Nylon carpet either. I don't know anything about Lifeproof to say one anything one way or another. I personally wouldn't put polyester in my house but that's just me (polyester is the least expensive and least durable carpet fiber).

My advice: go with SmartStrand or Nylon, it's hard to go wrong with either, but stay away from polyester, especially on stairs.

With a budget of $6.50/ft² for the whole package, you can look for carpet about $4.50 or lower. Depending upon where you live, $2 ft² for pad and installation (not including current carpet removal and disposal) is a pretty good rule of thumb. Much of the SmartStrand and nylon carpets will be more than $4.50 ft² but there are some out there. Ask a carpet dealer to see their carpets that are private labeled for them by the big boys. You can get pretty good value there if they have them.