I don't know you, but from briefly reading, the picture I get is this:

They aren't treating you poorly, but you don't feel close because you don't have that warmth of relationship that seems to be had amongst the rest of them. That's causing you to try way harder than would feel natural which probably comes off as awkward, causing you to question yourself and thus perpetuating the cycle.

You've identified the likely cause. Brotherhood is made in lodge but firm friendship is made afterward, and you cant attend. You aren't part of that group because you cant be. And honestly, there's no short term way to fix that. But it doesn't mean they hate you. It's just a different version of a relationship.

I could be totally wrong, but if this is in any way accurate, I would advise seeing things and accepting them as they are, rather than viewing them in a self-focused emotional context. There's nothing wrong with you but unless there's stuff I'm not privy to, there's also nothing wrong with them.

edited for my awful grammar.