It would seem you have a ASD individual next door, how you deal with this is up to your sensibilities, even a individual with ASD knows right from wrong, he should not peer over your fence, the fact that you told him this is very good.
You need to change his behaviour, you could try to use his sensitivity as a tool to do this, i.e. put a lamp in your garden, during the day you could put a brightly coloured object in the garden and change the garden around regularly, ASD individuals hate change. When and if he asks why the lamp, then tell him directly, you first complained to me, so I moved the lamp. It will teach him his behaviour as consequences.
This bit might be hard to understand but here goes, ASD individuals view people as objects, you may be aware of this in your lovely daughter, so the puppy will be of big interest to him and not the girl playing with it, sorry about that.
Something else you might consider is that if he knows your daughter has Aspergers and he is dx'd (diagnosed) as Aspergers, he may think, how does she interact in the world and recognise the similarities in her and his behaviour towards animals. This would facinate a Aspie.
If this is the case he will not know how to open a conversation relating to ASD.
When you are comfortable with his behaviour, then open a dialogue with his wife, and get the facts on his behaviour.
I think the fact that you are willing to be tolerant to him makes you one cool individual.
But don't put up with bad behaviours from anyone, there is no point in scolding ASD individuals or praising them it means nothing, but actions and consequences in a matter of fact way will get a response from him.
Good luck, I hope this helps.