As you are not applying at a normal transition point via normal admissions, you don’t have to live in catchment to be given a place if there is one.
On the other hand, if there is a waiting list and ‘in catchment’ is something that gives priority, then moving into catchment will move you up the waiting list, but not give you a place.
To give you an example, when we moved a while ago, there was a space in DS’s year which we applied for, and got, from about 50 miles away. I applied for a place for DD to start Reception, but as she was not an on time application, she was placed on the waiting list, way down because of the distance.
DS had 2 weeks to take the place up, and we moved into a rented house on the day he started. On the day he started, DD became a sibling, and took a huge step up the waiting list.
Our new address was verified by the Council, and then DD moved almost to the top of the waiting list, as a sibling very close to the school.
3 days before the end if the Summer term, the waiting list moved and she got her place.
So you may not need your ds / dh to move IF there is a place available in his year at a better school anywhere. If there isn’t, you can improve the chances if a waiting list place by moving BUT this is not guaranteed because nobody may leave or others (siblings, LAC, those with EHCPs) may leapfrog him in the waiting list.