The program at Disney is called Rider Swap. If it is only a 2 seat ride, then you would go to the CM, explain your DH cannot medically ride, and I would think they would give you a Rider Swap pass. This will allow you to ride with child #1 (while child #2 waits with your DH), and then you would use the swap to return to the ride with child #2.
Usually speaking, RS is for when there's a child too short, but if Disney won't let child #2 ride alone, it's only a 2 seat across attraction, and your DH cannot medically ride, then I would think they would work with you. If all of you can fit in one car on a ride, I would expect that Disney would have you do that and then not offer the swap in those instances.
ETA: If child #2 is old enough to ride alone according to Disney, but it's the parent that is apprehensive about the child riding alone, I doubt Disney would offer a swap accommodation, as they would allow the child to ride alone.