Was he unhappy?
Here's the thing about the fanatically devout: they are in it for the agony and the ecstasy. It's their compulsion, just like a drug or sex addict. Once they tell themselves that their god commands them to do something, they do it and do it until they achieve that cherished high. And when the rest of the world shuns and/or persecutes them for their words/deeds, they love it even more -- being a martyr is right up there with heroin and double penetration.
Considering that 90% of the WBC consisted of Phelps relatives, I think that Freddy was fairly programmed to be who he was at a very early age, and was probably happy with his lot for lack of exposure to any other kind of life. He might have had hidden urges that he tried to stamp out with his gay-hating, but that's just another symptom of the god addiction. If he couldn't get plowed by the boy next store, he'll just take it up the heart from Jesus.
I guess that at some point, each of the Phelps start questioning things, because several of his kids have fled the church. But I'm thinking not enough of them manage to get to that point in their heads -- they love the high too much.