I promised I'd give this some more attention on why I disagree about Bullseye, so here goes:
I fail to understand why superhero costumes are perfectly acceptable but supervillain costumes are not. I'm not saying they have to be crazy over the top and the show clearly had a reluctance for Daredevil too, but we're talking about Daredevil season three at the earliest. At that point, I hope they can be more accepting. There are a few things in favor of Bullseye being in costume:
First is the idea from the Dark Knight trilogy of the superhero causing escalation. Heroes wearing costumes can inspire villains to do the same for villains. This is especially true of someone like Bullseye, who has a bit of crazy on him. He channels that crazy into being a ruthless killer, but he's a showboat who loves attention.
Second is exactly that. He acts with style. He's got a swagger that I think works very well.
Here is an example of an image that shows this well. With the right actor, the costume would fit right in.
Third, this is a character used heavily by Frank Miller - a Daredevil writer who avoided anything he thought was cheesy like the plague. Sure, that was by comic book standards, but we're talking about someone who would only do anything over the top if it involved ninjas. Otherwise, he wanted it very grounded.
Don't get me wrong, I'd change it a bit. I'd make it black, not blue. I'd ditch the white belt and would be willing to ditch the white shoes. The one thing I would not touch is the white gloves. I think he comes off as visually distinctive wearing pristine white gloves covering the most dangerous hands in the world. I would actually have the first couple times he appears, keep him in the shadows and only show his gloved hands. Also, show the aftermath, like the scene in Guardian Devil with just a massacre of being having been killed by objects like forks and knives. Then you can show him in shadow with only the bullseye visible on his head. Finally, you can show him in full, but there's nothing wrong with it being at night since that's when Daredevil is active anyway.
Like I said, minimal changes will work wonders with the right approach and right actor. No matter what, I don't want anything resembling the Daredevil movie costume, though.