What is worse for my dog?
And that's the thing about outdoor dogs IMO - it's a lot harder to give them adequate care and attention than a dog who is inside and involved with your life constantly. So as long as you can keep the time commitment I wouldn't rehome him.
Is he having any issues that you can see from the current setup? How's the climate where you are (i.e. winter temps and snow)?
So does he sleep inside, since you said his crate is in the house? Making him sleep outside overnight in the winter would not be OK. Pits just do not have the coat for it, unless you are in south Florida or similar climate. And they really, really love people and need human contact a lot (as I'm sure you have noticed with him!) As a PP said, it's hard to tell from your post just how much time and when he's able to come inside. If you are able to give him lots of attention when he is allowed in, and go out to spend time with him throwing a ball, taking a walk once or twice a day, then I think you should keep him. Rehoming him would probably not be an easy proposition anyway. Pits have a hard time finding a home with someone who will really love them and have their best interests at heart. Do your best to spend time with him and give him attention, and get through this 6-12 month period.
I wanted to add that I have had several Pits through the last 20 years. Before the dogs I have now, I had a pair of red-nosed Pits that were my babies before I had human babies, you know? I had them from puppies until they died at 15 and 13.5 years of age. But there was a period, when my husband and I were divorcing. I moved to a distant apartment, and they didn't live with me. They stayed at his house, and he was often gone. They were alone a LOT (they had a dog door so they could go outside, but no humans in the house except for a roommate who fed them before and after work). I hardly ever saw them. They lived like this for more than a year until I moved to a house. And then they came to live with me again. The point is, they made it through the horrible, tough period, even without much attention, and then lived the last few years of their lives happily with me again. Your dog at least gets attention from you every day, and I think you can make the best of this less-than-ideal situation for him. :)