We have two rescue cats taken on in very similar circumstances to what the OP describes. Previously we have had rescue dogs too (all passed away at ripe old ages in our care).
It was quite apparent after a few days that the cats did not give a flying toupee about the change of owner slaves.
They do what cats do, they eat, sleep, play, fur up the furniture and kill the odd mouse mixed with an occasional show of affection to keep us on board and encourage us not to make them into cat-o-van.
A week of disruption while it settles into a new home vs. the current lifestyle for the cat I’d say is an ok trade off.
You may have made a mistake adding the dog but if something has to change then stick with what’s right for you but also consider actually which animal might be more adaptable.
It sounds like one of the two has to go.
Therefore one of them is about to get disruption to its life. The dog has been through that more recently and may in fact suffer more depending on its previous rehoming history.
Neither animal chose the current position – you created it (no blame implied here just a statement of fact that you/the family chose to add the dog) so disrupting the dog’s life on a last in first out basis isn’t really logical although it is an understandable emotional reaction.
Fwiw, Cats Protection helped our cats’ former owner and we were able to move from house to house without a cattery in between.
As an aside one of our two was known to have stress based UTIs, exacerbated by the dog. No signs of one of those six months later. Happier cat.