A Class A push-pull amp still needs a phase inverter.
It's a *single-ended* amp that doesn't. Not the same thing...! All single-ended amps are Class A - or they would be if they were biased to the optimum operating point, which most single-ended guitar amps aren't - but not all Class A amps are single-ended. (Nor is Class A the same as cathode biased, contrary to popular guitar amp myth.)
Confused? You will be if you read a lot of the nonsense about amp classes that's commonly used for marketing purposes!
Anyway, the phase inverter is a critical stage in the tone of an amp, but in most amps - including all Mesas unless the new Transatlantic is different, which it may be - you can't directly control the distortion that comes from it. What it mostly does is affects the clarity and detail in the sound, and to some extent the clean headroom. And contrary to what a lot of tube sellers may tell you, you do *not* need a 'matched' or 'balanced' phase inverter tube. It serves no purpose in the traditional guitar amp phase inverter circuit, since the circuit itself is not balanced... it just costs more.