You do not sound too happy with this situation already, and I cannot blame you. And I am assuming there is no way you can stay with your Mom either.
As hard as it would be for me, I would have to find someone to take the dog that would allow him to love inside and get all the attention he is accustomed to. It has been my experience, in watching different people with dogs, those who attempt this end up with a very unhappy dog and sometimes, without one as their dog runs away.
I know this is not your choice, so please do not think I am chastising you for something you cannot help, but I do want you to understand the way the dog may see this. It would be a lot like an abandonment issue for him. And the temperature change will be a definite problem.
Not knowing the circumstances that you are moving under, but assuming you do not have a lot of choice in the matter: I would first talk to my dad and explain to him exactly what the dog means to me. I would ask him his concerns about having the dog inside and then try my best to set him at ease. I would also ask him if he would please give it a month or two and see if he can adjust (and hope the dog can win him over in that time period). If not, I would find someone to take the dog after searching and making sure the dog was going to a good home.
As I said, I do not know your circumstances or your age, but based on what is going on, you may actually be able to get someone to take the dog and care for it for you if you offer to pay for all care. Perhaps check with some rescues and see if they know of any fosters who may be interested in doing this, and you could go and visit your dog.
If you will have to do this for a long time, you probably would want to consider re-homing because the family and the dog will become really attached to each other. Of course, the choice has to be yours, and I do understand how I would feel if I were in this situation. I was in it for a very brief time when I was younger (but not young enough not to be on my own).
My dad had a terrible aversion to dogs on the house from the time I can remember, even though he was very good with dogs as far as working with them and giving them very good medical care. In later years I learned why he was this way. He had lost a dog at a very young age and it left a terrible memory with him that he could not get past. I believe he regretted his decision when he was older and my mom got a small dog and just finally told him, I am keeping the dog inside with me! He got very attached and when the dog died at 16 years old he tearfully said that was why he did not want a $@#^ dog. But there was another dog there, and inside, in about two months.
So talk with your dad first. One word of warning: if he does agree to letting the dog stay inside, make sure the dog is not maltreated in that time. I would also watch if you decide to keep the dog and leave it outside. There are some people who just do not like dogs and will not treat them well. I am not saying this is how your dad would be. It is just something I have seen happen before.
Good Luck with whatever you decide and whatever you do. As I said, I do understand the situation. As for me, I found someone to keep my dog for a month and found a house on my own as quickly as I could. It took me two weeks to find the house and another two weeks to get it ready to move in. Separation from my dog was a great motivator!
Add: I just read your response. Believe me, I understand. My inclination, in your shoes as I see you are 17 (checked your profile), would be to get a small heating unit and maybe an air conditioner for summer and set up shop in the shed with the dog. I know that sounds drastic. But you are not going to be happy with the dog outside and he is not going to be happy there either. He needs to be with you. That is an adorable dog!!!
I would also consider the foster idea, if there is the chance you can move out soon and be on your own. There are people who have lots of love to give a dog, and are willing to work with someone who is willing to provide food and medical needs. Make sure you have a contract drawn up and signed if you do this and that it is not for an extended time. Have an end date in mond.
Good Luck. I will keep you and your dog in my thoughts. I know you love him. As hard as it would be, I would have to consider his health and well-being here though. Sometimes loving someone or something means loving it enough to let it go. But I would make it as temporary as possible. Like I said...sleeping bag here I come! :D