No kidding! I tried to appeal for reason and logic earlier, but my post quickly got buried.
What duty does a father have to support the beliefs of his child? We must seperate the two issues here: his daughter's comments regarding salvation and hell, and what to do on his parental visits.
(Background on me: I was raised in a conservative non-evangelical Christian house, had a Mormon babysitter, went to a Baptist school so conservative they thought the Southern Baptists convention was too liberal, and after flirting with Islam am a liberal Christian. My husband is a nominally Christian agnostic.)
First, at thirteen, the daughter needs to be taught what is appropriate in speaking with others about religion. "You're going to hell" is inappropriate to say to anyone, at any age, in any forum, regardless of how true the speaker thinks it is. He needs to get this across. That comment is childish, immature, and goes against everything Christianity stands for. "For God sent not his son into the world to condemn the world, but through Him, the world might be saved." John 3:17 KJV When I was about 8 I told me fellow babysittees that they weren't going to go to heaven. My babysitter (Mormon) told me flat out that saying that to anyone is rude. Everyone has the right to decide their beliefs, and insulting people will not convert them.
Second, he needs to inform himself about Christianity. Armed with a knowledge of what is and isn't in the Bible, he can have a serious discussion beyond, "It's all made up!" "No it's not!" et cetera. The Bible is full of mysteries. She most likely will not learn about them from her denomination, whatever it is. He will not learn about them by treating the book as a mind melting toxin. A way for to engage his daughter in a deep philosophical discuss would be ask questions about a topic (e.g. treatment of strangers), using the Bible as a jumping off point. He could point out the similarities in various religious traditions and secular thought.
Both should be forced to actually have a rational discussion. Reason and faith are not mutual enemies, and the irrational are of all denominations, faiths, and nonfaiths. He will never win if his goal is to switch her from an irrational Christianity to an irrational secularism. But if he wants to demonstrate that others can have different beliefs and still be good people, he can suceed.
So how should he spend his custodial weekends? First, he needs to acknowledge just how important her faith is in her life. Perhaps he could drop her off earlier so that she could attend a Sunday evening service. They could spend some time every weekend having a philosophical discussion and exploring where they agree and where they disagree. (which undoubtably will be an eyeopener for both.) They could visit other houses of faith and learn about what others say about God.
His time with his daughter is limited, both in the duration of visits and the number of years he may be able to have them. If he tries to spend his time making her choose between him and God, he's going to miss her.