The community here and on the Steam forums have discovered the official unlock conditions for Termi-Nation Bunker!
Defeat Omnitron in Omnitron-IV with regular Bunker as the first hero. Then, defeat Cosmic Omnitron in Omnitron-IV with regular Bunker as the last hero. At the end of both games, Bunker must be active and have less than 10 HP.
YAY!
Funny that the unlock that was the last to be gotten/replicated was the first one we got the exact conditions for.
Now that we know the unlock requirements can some one explain them from a story perspective? They make no sense to me. I don’t see any special relatioship between Bunker and Omnitron mentioned anywhere and never heard anything on the letters pod casts that would remotely lead me to figuring this out.
Look at the incap side of the original version of TermiNation Bunker. It also got mentioned during the Omnitron Letters Page episode that Omnitron IV used pieces of TermiNation Bunker's suit that got damaged fighting Chokepoint.
Wonderful that yall figured this out. But why oh why did it have to be Omnitron? He's the one "easy" villain I consistently struggle with. Baron Blade, yawn. Ambuscade, training dummy. Gloomweaver, seriously? Looks like its finally time for me to figure out how to beat him. Damn you Sedative Flechettes and Technological Singularity!!
My team
Legacy, KNYFE, TL Tachyon, Wraith and Bunker.
Tachyon can get all the cards and can stop damage
Wraith can deal the damage (with all the boosts) and has decent control
Legacy, well, it's pretty obvious.
KNYFE i chose because her damage and play is pretty intense.
With this team I destroyed Omnitron 1 in a few rounds. Same team took a bit lonegr for Onitron II, but still sucess.
My team, besides Bunker, was TLT, Dark Visionary, Parse, Argent Adept, in that order. Parse can prevent Flechettes or Singularity, Argent does his thing, Tachyon helps as usual, and Visionary helps control things as well as cheese it out with the EPE (I did this both times).
Bunker should focus on spamming his adhesive foam grenades, whittling the boss’s health down, as well as his own; get him below 10, then make sure to end the game STAT.
With this team, it’s a surprisingly easy set of fights.
I used a variation on this – substituted regular Legacy for Argent Adept (for tanking and increased damage), but otherwise the team was the same – with similar results. Got the unlock in one try.
In theory, at least. I had a game recently where Parse noped the Fletchettes only to replace them with more Fletchettes.
Ouchies
I guess your plan wasn't fleshed out! HAHAHAH! 
Yeah, sometimes the deck just decides it hates you. Although if you are also using Extra Sensory Perception. You should be able to seriously mitigate the chances of that happening.
I've got worse; trying to get the unlock I had two Buffer Overflows out, and the enivronment played a drone which got Cosmic Omitron started. I sent away Fletchettes to get a fatal EPE, and sent that away to get the OTHER fatal EPE.
That’s what Visionary is for! The EPE is your friend!
Got the unlock. First game was easy with Bunker, Legacy, TLT, Wraith and OX. Second game was same team with OX and Bunker swapped (and cheesing the rewind function with the Infrared Eyepiece).
Ended up having to have Bunker shoot himself a couple times to get under 10 HP in both games.
Not if the EPE is played by the environment turn.
That's why I brought OX – never let Cosmitron flip in the first place.
Oh, the ENVIRONMENT did. I missed that.
I had a few bad runs against Omni 2. The worst was Legacy, Fanatic, KNYFE, Tempest, and Bunker. Everything was going well, but we were a little hurt. I finished off Omni 2 with KNYFE's Incidental Contact, but had accidentally healed Bunker to 11 with Cleansing Downpour. The worst is I could have hit Bunker first and gotten the unlock, I just wasn't paying attention.
Got it. Omni I was pretty easy with the team suggested above thanks to Parse denial card eliminating Fletchettes and 3 Critical Multiplers and that environment card that let’s you discard 7 plus cards to destroy it.
Cosmitron though took 2 tries. Couldn’t get Parse’s denial card at all both times and Bunker got taken out by an ill timed Technological Singularity the first time through. Oh how I hate Omnitron.