My advice is to ride it out. Your son is learning to deal with a new situation. No longer does he have one home with one bedroom. He now has two homes and two bedrooms. That will take some adjustment. It sounds like he is rebelling against staying with Dad overnight because his is the new house, new room and not the one where he spends the majority of his time.
I know as a Mom of 3 of my own children that your are concerned with his mental well-being and happiness. But I have to agree that riding out this aggravation in the short-term will make him happier and healthier in the long term. I think this for several reasons. One it is forcing your son to learn to adapt to new situations and changes in his life (which will always occur. some small, some large but there are always changes.). Also, it will allow your ex to maintain and develop a better relationship with your son and your son will come to understand that even with the changes his Dad is still there for him and still loves him. Next, having a 3 1/2 year old myself, I know that sometimes the rebelling against staying in a certain place or with a certain person is just a cry out that the child does not like the changes going on and doesn't want to deal with them. Although I am not divorced, my 3 1/2 year old is dealing with having a new younger brother (now 7 months old). Different situations but both major changes. And, of course, similar reactions. Mine doesn't want to sleep in his room because Mommy isn't there holding him while he goes to sleep. Instead, she is feeding and taking care of his little brother. He doesn't have a problem with the jealousy issues during the day. They usually only crop up at bedtime when Dad is putting him to bed.
Also, if you and your ex have agreed upon certain bedtimes and stick to them, I don't think you will have a problem with his sleep patterns in the long run. You may have a little problems with being overly tired the day after he has fought Dad about going to bed for a couple of weeks (since he is only there once a week). But if you and Dad both stick to your guns about the same bedtime, that and your sons arguments should lesson as his schedule becomes more predictable and routine to him.
Good luck with both your little guy and your ex. I H. my advice helps you out.