Okay, I can finally add to the list and answer my own questions. Today, I put my rifle together. Here are the parts as fallows:
Post 2014 AICS AX chassis green
Bighorn TL3 with all 3 bolt heads, Thick tang
Huber trigger
223 Proof barrel pre fit
260 Remington Bartlein
300 WSM Bartlein
AICS 308 mag
Ruger 223 mag
Accurate 223 mag
AICS AX magazine
AW magazine
AX magazine
AI 223 magazines on the way will update
My TL3 is cut for an AW magazine. Upon installation, I could not get s single magazine to latch into the well. After looking at it, I had to remove the pistol grip to install the action and make a small mod just under the bolt handle. Then I re-installed the action and torqued down the action. Not all mags were sitting where they are supposed to. After chambering the 260 and 300WSM, I installed the 223 Proof barrel and loaded some rounds into the mags. All 10 cycled from each magazine. I will try the AI 223 mags as soon as they arrive and update.
I swapped out barrels and bolt heads and tried the other mags. The AW, AX, and AICS AX all fed perfectly. For the 300 WSM, I had to mod the feed lips on the AICS to get it to work, but now feeds well.
So a long story short, I think the TL3 can use any magazine with the AW cut that I have in the post 2014 AX chassis. All the magazines I used fit well in my action/stock combo. All were snug, not loose like my other rifles with a M5 installed. I checked the side to side also and it didn't seem much on my rifle.
Thank you everyone for contributing to this thread. It was very helpful because I needed a plan B If A did not work.