I'm pretty sure they all believe, with the possible exception of Fred himself. Sure, they're also living off the lawsuits, but that's not incompatible. Fred was a lawyer, he had all of his kids become lawyers, because he sees it as a way to get money from the unsaved. And why not? To the Phelpses, we're barely human. It's perfectly all right to get money from us. If he could find a convenient way to just take the money, he would. Since the only legal way he can get it is to provoke people until they can be sued, that's what they're going for.
It's a lot like white supremacists who complain about black welfare queens while living off government handouts themselves. They see themselves as worthy of that money. They're good people, by virtue of being of the right religion/race, and they're taking it from bad people (who keep them from attacking/killing the groups they hate), so it's morally justified.