"Is BAT regarded as a notch below QUAD?"
Nope. But what do you care? It's how you respond to the sound that counts. You're lucky because you have been able to listen and compare in the same circumstances. If you have not been able to make up your mind yet, you haven't let your musical intuition work on the decision long enough.
Get in some more listening time and pay attention to your emotional and physical response, not to sonic characteristics. Your body will tell you what it likes best: tappy toes, the urge to dance, a feeling in the heart or the stomach... something like that.
For my 2 cents, you already know which one moves you most. Trust that.