Never had an AT-1 but had a Tone Zone and still have a Breed and two Super Distortions. What I hear as common denominator in these pickups is strong bass and lower mids, very high output, strong compression and quite a lot of fizz on top. They sound good in some guitars but painfully grating in others IMHO. None of the BKPs I have tried sound remotely similar.
The Air Norton is a different animal. I have tested it in both the neck and bridge spots of different guitars, and it always has worked well. Bass is not too strong which helps it to stay clear as a neck pickup - it is indeed remarkably clear for such a high output in the neck. In the bridge it is strong with a strong center and high mid focus. Sounded good in my LP until I changed the bridge - then it sounded nasal (no I didn't change the bridge back because the new bridge had to stay to save the neck pickup). It sounded great in my MIJ Charvel. It's not fizzy on top. It's definitely my favorite DiMarzio pickup, even though it is not in any of my guitars at the moment.
However, based on the reported spec the neck pickup in the axis is not that close to an Air Norton which measures only about 12.5k DC resistance. If you are going by DC resistance and other specs for the axis neck pickup then a Rebel Yell bridge seems the closest to me. Strange - I would never think of putting a contemporary BKP bridge humbucker in the neck spot.
Cheers Stephan