My first thought, “That is a terrible beam, looks like something other than TEM00”.
We would like to see the beam on m1 with something that actually shows the beam.
You don’t have an option here, really. If the beam is coincident close and far, that is where the beam is properly adjusted. You need to ‘catch’ the beam with m3. Meaning it has to be adjusted in both the Y and Z directions.
This is a video from Russ Sadler that is a good tact to alignment. He is using the lightweight head he designed which simplifies this. The idea and technique will help you out, I think…