So you are saying Fallout 3 isn't canon? The call themselves the Enclave alot in Fallout 3. I agree Fallout 3 has bad writing, but in this case it is Bethesda saying "They care the Enclave." So therefore they were also calling themselves the Enclave in Fallout 2.
They were calling themselves the Enclave in Fallout 2 anyway:
"You live so that the Enclave will survive."
I don't take that to mean anything more than that, Enclave is just an interchangable term taken on. Of course it was the name that the Shadow Government had on their business cards, they just then started calling themselves the U.S. Government just because I guess. In any event Eden still refers to himself as both President of the United States and that the Enclave is the United States Government.
Does he say he hates the Enclave? or are you just putting words in his mouth? Maxson killed the scientists at Mariposa because he Hated what they were doing. Therefore he hated what the governemnt for which he served were doing, therefore (by your logic) we can't trust anything about the Brotherhood.