It's not logical to expect someone to only have sex in the morning, and frankly it sounds like an excuse, particularly when you say that you aren't having sex in the morning despite his claim. Rejection does hurt, and he's not even going through the motions, he's just rolling over. Something isn't right, but I assure you that blaming this on yourself is the wrong way to go. First of all you have to assume that this has nothing to do with you, you're not asking too much, its not because of your weight, it's not about you.
If he's always been this way, then the fact that you had no privacy has always been just an excuse for his low sex drive. It wasn't until you moved in together that you can see that it was never about privacy. I don't know about him being gay, but despite his protests, it's a possibility, just one that he may not be able to ace. He may not even know consciously. It could be something that happened to him when he was a child, or something that happened in his past relationships that is contributing to it, but the fact is, unless he gets some type of help and is willing to open himself up to that help, then things will never get better.
You're going to have to decide if this is something you are willing to live with. I welcome your thoughts, let me know if you want to talk more