@Fressia123
It's a problem in the sense of a "mathematical problem" not that he's a problem. He's a lovely young man and I wish I could give him a decent alternative that he's happy with.
Only I WFH and there's also a cat too
He gets the dining room as his own room. You work from the pod in your garden as do many, many other people who work from home.
I have a folding dining room table which fits four chairs inside it. When it is not in use, it is a slender side table. You could get one of these for your lounge if a dining room table is important to you.
Why are these solutions not obvious? You talk about wasted space in the house on the days the children aren't there, but if you won't let your dss sleep on a sofabed or allow your baby to share a room, converting the dining room to a bedroom would seem to be the natural solution.
You bought a house knowing it was too small for your needs (four kids, growing family, wfh, musical instruments) because you didn't want to get rid of your dog to rent somewhere bigger. That's a choice, and I understand why you didn't want to get rid of the dog, but this means that sacrifices need to be made elsewhere. Some sacrifices are better than others.
An attic extension would be the ideal solution if you can make that happen - hopefully the architect can give advice.
Sending a 13(!!!) year old boy to sleep in the garden every weekend in a structure with no plumbing and limited electricity is not a sensible alternative and not something a parent should be seriously considering in 21st Century Britain.
Basically you are on here saying that it is more important to have your computers and piano inside the house than a 13-year-old child. Do you really not see how mad that sounds?