Does FAA simulator (AATD/BATD) count toward experimental rating 1,000 hours requirement?
The short answer is no. As you said in your question, the regulations you quoted require you to have logged flight time, which is defined in 14 CFR 1.1:


  Flight time means:
  
  (1) Pilot time that commences when an aircraft moves under its own
  power for the purpose of flight and ends when the aircraft comes to
  rest after landing; or


In other words, you have to go fly a real aircraft. And 1.1 defines "aircraft" clearly too, so you can't argue that it includes simulators.

But even if you could use a simulator, you would still need an instructor present. 14 CFR 61.51 talks about logging simulator time for various purposes, but in every case it also says that an "authorized instructor" must be present and endorse your logbook. That makes sense: if you're playing around by yourself in a simulator, you could be doing anything at all, including reinforcing your own bad habits. The instructor is there to ensure that what you're doing is useful and relevant, and you're learning from it.