Ok, lot's to comment on lol! Let's see if I can try to include everything and not make any repeat suggestions.
First, let me say that I absolutely love your son's name. I usually don't like double middle names, but Archer Cohen Bear has the perfect flow, and it's so handsome. Cohen is a name that is very high on my list right now. I see how it would be so hard to find a name to go well with his name. I agree that you need a middle name that is equally unique and with the same type of feel as Cohen Bear.
From your initial list, I think Roman, Brody and Gavin work very well as first names that would go well with Archer. I've never really been a fan of Everett, but there's absolutely no problem with it. I just think the name John doesn't compare at all with Cohen Bear. Have you considered the name Jensen for a middle name, like Everett Jensen? Gavin Ezekiel is great, and would flow with Archer Cohen Bear, seeing as Ezekiel is pretty uncommon, but sounds really cool.
From some of the suggestions that you liked, I think Roman Silas or Roman Thatcher (even though you didn't think you'd use this) would be nice. I've always loved the name Roman, but I also completely understand what you mean about names losing their freshness. I've had my lists for so long, and have been adding and deleting and altering, but I still don't have any that scream out at me. Good think I'm not expecting just yet!
A few others that I put together from your list:
Ezra Flynn (although I'm not sure about Archer and Ezra as a sibset)
(By the way, some of my favorite names are Caius, Edison, Silas, Colton, and Callum)
And finally, some more suggestions (as if you need anymore!):
Some names that could be used as a strong, unique middle name to go with Cohen Bear:
Titus / Titan (on my GP list)
Kincade (recent name I fell in love with, also on my GP list)
Braven / Bravery (maybe a little too literal, but they are on my GP list)
Artemis (but maybe too close to Archer)
Knightley (another GP)
Thorin (possibly a little too similar to Thor though, lol)
Whew! Ok, so I hope my opinions helped a bit! I have a feeling you'll end up choosing a great name, considering Archer Cohen Bear is so great.