If it were me, I would ask him politely and then promise that you will do something that he doesn't like doing himself in return for his help. And make sure he knows that you appreciate all he does. If it feels like you are mutually helping and supporting each other in different ways and he also gains something from this, he might respond better.
Also, see about spraying your home for bugs - do a full spray around the walls, windows, doors, along the walls where they meet the ceiling and floor, etc. and if you buy a quality product like Ortho Home Defense it will create a perimeter and keep the spiders out of your home so your husband doesn't have to kill them individually all the time. Do see if your husband will help with this, though, as sometimes existing spiders will flee once they sense the poison and that might freak you out if you have to do it. Then do him an especially big favor in return for spraying.