Unpopular Opinion
I like the idea of Bullseye joining the Thunderbolts more than him staying has a Daredevil villain. I hear a lot of people saying that the MCU has a problem with keeping villains as villains but I think part of that issue is that they kill off a lot of their villains such as Ikaris, Gorr and Wenwu.
Also for a team of reformed villains the Thunderbolts in the MCU doesn't actually have the many actual villains, Yelena was only really a villain in the Hawkeye TV show and that's only due to being manipulated by Vanessa. Bucky was brainwashed as the winter soldier. Walker did kill a man who was surrendering but that man was a terrorist who just helped kill his best friend (not saying it was right just acknowledging it as his main crime). Red Guardian aided the red room so I understand why he seems like a villain in some ways. I haven't watched Ant man and the Wasp so I'm not aware of Ghosts crimes but I'm going to assume there are a lot. I'm not counting taskmaster since she's dead now and no longer on the team.
I think Bullseye would work on the team because he's actually trying to be better now but he still loves killing and shows no remorse for it. I like the idea of him being a Thunderbolt because erasing his "redemption" arc would just feel a bit boring and I don't just want to see him as a generic Daredevil villain. I think his Thunderbolt arc would function as good setup for a Lady Bullseye villain with him saving her while working for the US government but if not there are plenty of other good Daredevil villains Marvel could go with that keep Bullseye as a kind of anti-villain. Don't get me wrong Bullseye should never go full hero but anti hero could possibly work.
Anyway that's just my viewpoint and I'm happy to be challenged on it. :)