The Stiller MK13 action is a TAC 300 with a different engraving, the RFID tag and minor change in finishing from what I've read; I have one too. It comes with a stiller 20 MOA action rail and is drilled, however, in the AI MK13 AX chassis kit, there is a replacement action rail labeled Stiller TAC 300 20 MOA which fits the MK13 action, as you would expect and this gives you a same height contiguous rail between action rail and forend rail. Using a MK13 action in a Mod 5 would mean addressing the different mounting hole locations of the stiller MK13/Tac 300 action rail holes and the holes in the Badger/Remmy Mount.
What surprised me about the MK13 action is only the top rail has MOA, none of the side rails have MOA in them, I would expect you'd want a laser designator/PEQ on same plane, so my speculation is peqs would be run on the top of scope caps or on the top rail of the new set up, but just speculation on my part.