Tassiter gets killed by Jack in a cutscene after the main game credits, so using him as the main villain for a DLC would have been hard.
At least Tassiter was kind of a villain for part of the Claptastic Voyage story. (He hid the H-Source inside Claptrap, protected it with some security measures, and left the Override Key there as well.) Though I agree that the main game could have used some more Tassiter. Perhaps some comments during the story mission in the Titan Industrial Facility and Robot Production Plant maps. Like this: “You want to use a Dahl AI against Dahl troops, John? I doubt that’ll work.”, and before the Felicity Rampant fight: "Hahaha, that’s what I expected. I’ll enjoy watching you get your ass kicked."
Tycho’s Ribs and/or Eleseer would have been a great place for some more Tassiter sidequests. During early game it’s understandable that he only delivers some comments because he doesn’t take the vault hunters seriously, and Jack goes mostly radio-silent. When Jack, the vault hunters, and the robot army return to Helios Tassiter recognizes them as a danger to him, so he tries to divide them (see e.g. sidequest “Red, then Dead”), but after that he kind of just disappears until Jack kills him.
I think the side room with the many chests in Tycho’s Ribs would have been an easy grab for a sidequest. Put a bounty board at the start or at least somewhere before that room where a single sidequest is offered by Tassiter. (“Jack doesn’t deserve all that eridian stuff. I want some too, get it for me. I identified a location that is apparently a storage room. Loot it.”) Have him send the vault hunters into the enemy-infested room (“Oh, it’s guarded. Well, just shoot them. That’s the only thing you can do reasonably well anyway.”) and then the chest chamber, where you turn it in at the central chest. (“It’s just a storage site for Lost Legion weaponry? I have no use for that. Take them, it should be reward enough considering you just wasted my time.”) Reward would simply be a choice between two green guns.
Yes, it’s a short side quest and doesn’t explain much about Tassiter, but it fills the huge hole between his sidequests and messages on Helios and his death at Jack’s hands. If you put in one line per vault hunter (maybe as reaction to “you just wasted my time”) which ranges from calling him a douche to wanting him dead (depending on character), it’d nicely end the story arc about his relation to the vault hunters. Especially if all characters end with something like "Don’t call me again because you are just wasting my time."
As a nice side effect, it would break up the dullness of Tycho’s Ribs a bit more. Throw a miniboss into the room and Tycho’s Ribs would even become a map worth revisiting.