While it is true that some of children's stuff will sort itself out as they age, some will not. You only get chronological and physical maturity "for free" as they age with each birthday. Intellectual maturity, emotional maturity, social maturity, and philosophical maturity you have to work at. An Asperger patient child could have extra challenges there.
And teaching them those things and their willingness to learn and defer to your household rules gets harder as they get bigger/stronger. Dealing with a baby size person's tantrum is one thing. Dealing with an elementary school, middle school, or high school size person is another!
My dad is impaired with Alzheimer and he's a big guy at over 6 ft tall. When he was pitching a fit and throwing things about -- it was not pretty!
Your husband is willing to risk that kind of thing? The child growing to a physical size where it gets really dangerous? It's one thing not to like the current therapist. It's another to not give the child every chance possible.
I am so sorry you both cannot come to terms on his caregiving needs. But I really think this is the area you guys could spend more focus on right now than the threesome thing.