This is such a sensitive issue OP and I'm sorry you're getting absolutely flamed.
I'm going to add that having a house covered in cat wee and poo is not an inconvenience. When you have a baby that has started crawling, it is an absolute health hazard. A baby could pick up some very nasty life-threatening diseases from cat poo. Please could everyone remember that.
By the way, the comment upthread about NEVER owing a pet again if you dare to suggest PTS was unbelievably harsh. The OP has said that she loves the cat to bits. No pet owner would be so obviously heartbroken if they didn't care about animals. Maybe we could all have a bit of sympathy and compassion for the OP who is obviously stressed, scared and heartbroken.
Trying to get it rehomed is an option but who is going to want to take on a stressed cat with serious issues? Taking the cat to a shelter is another option, but it would live in a cage which seems cruel.
Banishing the cat to one easy-to-clean area of the house is a great idea, as long as it is roomy, has windows, a cat flap and hard floors. So, this is also not easy for a lot of people. Banishing it to the shed sounds really harsh but if you can do it, it might save your sanity. Maybe the cat could come in in the evenings for a cuddle once your DS has gone to bed.
We have a cat. She was our baby before DS came along. She has always had stress issues and regularly goes through phases of weeing and pooing absolutely everywhere. DS once crawled in some and got it everywhere. Literally, everywhere. It was disgusting. We have all stood in her urine and poo and tracked it all over the house on more than one occasion.
I wrote a post on here a few years ago which was almost identical to yours OP. I dared to suggest that I was at my wit's end, and people absolutely flamed me for thinking I meant I was going to PTS.
Anyway, our cat has got better. We have made one major change which is that the cat is put outside at night or if we leave the house for any length of time. Flame away if you want to. I could not cope one day longer with coming downstairs EVERY morning to find a big pile of sloppy poo (digestive issues) and a huge puddle of wee. Every single morning. So she goes outside. At first I had sleepless nights because I felt so guilty. Now it's totally normal, the cat is fine, and the house is much improved. The downside is that all the paths, grass, patio... everywhere is covered in sloppy cat poo and it smells disgusting. But at least it's not the carpets.
OP can you try that? Put the cat out at night and when you go out. Keep an eye on him in the day. Watch his routine like a hawk. Our cat wakes up around midday, eats a tiny bit, has a poo, then eats some more. Like clockwork. She does the same later in the day. I have everything crossed that this works out for you.
Blimey, that was long!