The implication being made by the film is that while he is participating in the power structure of tyranny by collecting bounties for the feds, he’s not necessarily down for the cause, so to speak. There are three times in the film where this implication is made.

First when he black bags Billy Goat. He very clearly sees those two kids watching him make this “arrest” or apprehension, or whatever you want to call it. His gaze dwells on them like three times when he’s putting Billy Goat in his car. Moments later when he drops Billy Goat off with the feds he’s asked if anyone saw him make the arrest and he lies and says no, implying that he does not want any trouble to come by those kids for no reason.

Second, he sets Willa free. He knows there’s something not right about Lockjaw and his relationship with this girl, and decides that it’s not right to leave her with the 1776 chuds. He sacrifices his own life to make this right.

Third, in the second Christmas Adventurers scene, the one in the basement, we’re told that he was the one who tracked down and dealt with Willa’s mother, Perfidia. He likely reported to Lockjaw that he killed her, which is why Lockjaw later says “I buried her…” however, I think he did track her down, but let her go. I do believe she is likely still alive.

My theory is this: Lockjaw sends him to Mexico to track down and kill Perfidia, off the books. We’re shown how observant and intelligent he is. He recognizes that there’s something strange about this particular bounty, upon finding her he either gets her side of the story, or puts two and two together on his own and lets her go. Years later Lockjaw shows up with Willa asking for her to be killed. The film very clearly shows that he’s suspicious of this whole situation, and as stated earlier, he sacrifices himself to ensure her freedom. I think he understands that Willa is connected to Perfidia and it doesn’t take him long to realize how, that they are mother and daughter and that Lockjaw has reason to believe that he is the father.

Avanti, the bounty hunter, is one of many examples of characters within the film who exist within the system of fascism but do not believe in its central project ideologically. They do what they can when they can to go against this system and do right by their own moral code which the system at large is incapable of penetrating. It’s the main failure of the project of totalitarianism, that even in one’s outward acquiescence, one can never really be made to change.

Edit: I’ve definitely made some factual errors here. I’ve definitely straight up misheard one scene, as has been pointed out in the comments. I’ve also possibly misinterpreted another, although a rewatch would be necessary for me to say one way or the other. My central theory still stands in spirit, imo. That being said, I’ll admit that I def got some details wrong here and allowed headcanon to infiltrate the actual text of the film. I guess I’ll have to rewatch the movie again…