If a believer's faith in a religion has become weaker, what is the word to describe this state? Can I say "He has backslid"? If not, what is the correct word to use?
It's a good question. I'm not familiar with a single word which summarizes a state of "weakening faith".
However, the terms Disenchanted and Disillusioned apply in a more general sense, and come the closest in meaning.
You could say, "He has become disenchanted with his religion", or "He is becoming disenchanted with his religion".