I would close the door. Depending on the size of the crate, I don't know if I would put a pee pad in it. You don't want him to get used to going in his crate! If you are okay with getting up through the night, I would do that!
When we first got Angel I got up through the night with him. I had to wake him up!! When I took him for his first vet visit, he told me not to wake him if he is sleeping. So, I didn't! I'm mixed feelings on this topic, but he is 2 years old now and seems to be doing fine.
The play pen is probably the best way to go. You get the best of both worlds!