This is exactly one of the things that's nagging at me.  Sorry I have no insights yet.  I have to rewatch the episodes but this is one of the things I'll be paying attention to. 

The FBI badge has been significant since the first scene with the Giant.  They didn't repeat the scene like I thought they would, so we still are not sure when that happens.  If Cooper is "far away" we now know that could be literally anywhere, dimensionally speaking.  Cooper doesn't know about Richard and Linda until he wakes up in the hotel the next day, but when the Giant speaks to him about Richard and Linda he says "I understand."  So if he doesn't have the badge during the conversation but after he knows about Richard and Linda, WHEN does this take place???

I think we all need to rewatch them, and probably season 1 ,2 and FWWM too. I bet in time a lot more will start to get a little more clear, or at least less obtuse.

Thanks for the answer!

Nice spotted Filipe!

... Thinking about it, the badge also looks a lot like the thing Bad Coop is wearing in the back of his hair...

Thanks!

I checked about that but it seems that are not the same object.

I assume Mr. C was just some fashion enthusiast (?)