I don't see that. That's like saying radical evangelicals have always wanted to increase hatred against Christianity. The relation between the 2 is that one acts as a recruiting ground for the other.

Radical evangelicals like radical Islam think they are the "pure form" of their religion.

I don't see how you can make that claim. The only possible benefit I can think of for them would be that more bigotry towards Muslims might sway more people towards their point if view, but they were already in close contact with a population thoroughly abused by the West in Iraq.