Not really a comparison, your house is neither on fire nor are you forced between to choose the kid or the cat. The cat is not endangering the child, you only have listed the child as a reason for using interest (reason is not an excuse).
Your choice essentially breaks down to this:
You are in a normal building which is not on fire. The cat that your wife took responsibility for is stressed out and likely does not get any attention as interest has been lost. As a result of whatever distress your cat is under, it pees on the matt at night.
1. You can choose to keep the cat that your wife took responsibility for and spend a little effort accommodating it with a cat flap, relaxing feline spray, lil bit of attention and let it get use to the house.
2. You can choose to rehouse the cat, although it will likely stress it out some more
3. Lock it in the kitchen with a cat flap, which is cruel considering what it has been use to so far in its life
4.lock it outside which is incredibly cruel.
I really don't understand how any pet owner can choose any option but 1, especially if they have had the cat for years.