I can't see how the coils would help the aem from shutting down but its worth a try.
plugs for the tt ecu mod are the ones you want in general for a supra yes. if you are going north of 18 psi make sure you are at least in the 7 heatrange (so bkr7eIX or if alot more boost then drop to an 8 so Bkr8EIX), still wont cause or solve your issues.
fuel upgrade is always good but dont think its your issue.

consider there may also be something wrong with the aem, I just got a aem v1 from a moderator on supraforums and it looks like it wont turn my fuel pump on no matter what I change in the settings, so I think the "eprom" is "loose or bad" if you search online it happens to v1's alot and then weird things can happen. not saying its your case as you also have a v2 but may be worth sending in to aem for a lookover, or at least calling and asking they may have heard of the issue before (Although they aren't always going to admit it) mine will be going in this week the diagnostic fee isn't too crazy to be sure everything is ok with the unit.

 

Hmmmmm.......
I may have to send it in this week if we dont find anything on second dyno date. We are going to watch electrical stuff while it shuts down on the dyno friday after the fuel and coil upgrades that likely won't fix my issue anyway. I am doing them because It was in the plan anyway. I also hoped to get lucky.
Thank You Sir!

 

I have also been carrying a high output (200amp) alternator around in my trunk. Why not throw that in too, LOL.

 

It looks like the box may be the issue.