I grow in closets, it does get colder at night. As for it being a little colder I think that might actually help because the lights will be warming it some. What I am thinking of doing after this harvest is switching the 12/12 light schedule to run overnight. Maybe you could do this as well. I am going to see how it helps on there heating bill I can almost guarantee it will. The lights keep the back of my house warm I just have to heat the front but during the night both need warmed. It should also protect the plants from the cold radiating through the walls.
How tall is the closet before it gets to the start of the angle in the roof? I don't think you could hang a light from the bar since it is not centered but you could use it for hanging electrical cords. I would just get some decent eye hooks if the roof is secure enough.
Is there any space above/around the door? If you had the room you could put a 4 inch intake fan and let the air exit through cracks. I do this with my veg box it stays a decent temp. Could also get a small fan on the inside to circulate the air.