A few observations, in no particular order:
Isn't it a no-no to have the service conductors and feeder conductors in the same enclosure and/or raceway?
The neutral/ground bond strap in the panel should be removed, and the green jumper replaced in the ATS.
The generator-to-ATS feeder conductors need only be sized for the generator breaker, if I'm not mistaken.
I can't speak to having them in the same raceway with certainty but a quick search does seem to indicate as much. I am planning to replace the SER soon so when I do they will be routed from the attic soffit directly to the ATS. Since it's not a direct safety issue ( no one will be in the panel except me) I'm going to let it ride for the next month or so until I do the replacement.
This does bring a question up though- When I do this my SE will now be outdoor (surface mount conduit from attic soffit on exterior of wall. NOT underground) so is there any issue with this application?
The Service panel also is my primary, first in line from the meter, service disconnect so the Neutral and ground should be tied together should they not? Therefore would that not make the ATS technically a subpanel and therefore have isolated ground & neutral?
The #3 I got from table 310 as well as this forum based on a 100 amp (generator breaker size) service at 75º. And it also is what Generac sells for the 22kw if you use their bundled wire. The run is roughly 25' from ATS to Gen.
Roger is correct that we are still under the 2014 code.
This is exactly why I posted so I appreciate all of the feedback. These forums are a gem for knowledge especially for us that don't do this every day.