Do you think it's a case of changing the bias point of the AX7/ AU7?
No.
It has more to do with output headroom. Champ is simple SE 6V6. Switching to Au7 does keep things cleaner only because you are losing gain. The limits of the output stage are already set. Just guessing here but like most all Fender amps I would assume that Fender designed the Champ to be as clean as possible for the given circuit.
But yes, a SE 6V6 can be ran somewhat clean but only at very low volumes - much lower than most guitarists are accustomed to using them at. But still call that semi-clean when compared to a Twin Reverb. If just for recording this may work fine for your needs.
You may want to consider the 5E3 Deluxe (PP 6V6). Yes, they are know for dirty gnarly Neil Young tones but like a Champ will be somewhat clean at very low volumes. The bigger speaker and higher power will afford slightly better volume in the cleaner range.
The one thing that will help greatly in keeping a Champ or Deluxe clean sounding under higher volume conditions is by using a BIG PA type speaker like a JBL or near equivalent such as the Weber High Powered Ceramic series. Also, avoid OT saturation by using a hefty OT. Some people purposely use a small core OT for distortion effects. The original Champ OT is probably midway between oversized and undersized.