I would do the same, take him out at 5:10 but don't talk to him, no food or water, no praise or treats, one minute to pee if that's the issue then back in the crate till six. He'll protest a bit but eventually get the idea that it's not morning time yet. Putting some background noise in and blocking the sun will help too, if he's hearing traffic or seeing light. Not the instant solution but at least you can snooze a bit longer that way. Keep the five am pee break quick and boring!