Overall, hesitant to potentially have a crowing rooster at 2 am (or early 2,3,4 am hours) through the winter. I do have neighbors and don't particularly want to be the subject of complaints. In addition, I have a neighbor with a rooster that crows more often than ours. Their rooster, and ours respond to each other...so if mine crows, theirs might as well, and so on and so forth.
Due to the natural light waning in the evening, the birds have been settled on their roosts before the light turns off.
Don't worry, roosters will naturally crow at 2:00AM if and when the notion hits them. Maybe you should concentrate on creating a sound proof or at least a sound resistant coop to hold a loudmouth rooster. There is a surgical procedure that will quiet a rooster and if he survives it (and even if he doesn't survive the procedure) his crow will be quieted. It involves cauterizing the roosters vocal cords with a red hot steel wire.