Hi, my DD has just turned 3 and we've been dealing with this for about 6 months. She usually only does one or two a day but they can be awful, I know how stressful it is constantly changing/washing/chucking out clothes!
Having tried to keep tabs on what she's eating, the only thing I've noticed making it worse is dairy. We've cut back on things like fruit juice/smoothies, and she eats a lot of white bread in an attempt to balance things out a bit. I think when her diet is plainer things are a bit calmer, ie we have one stool a day that is still 'toddler diarrhea' but doesn't manage to escape her nappy.
I do feel for you though, we had to stop her ballet class because it was no parents allowed and I didn't feel I could leave her in case she had an accident. We still take her swimming but not weekly, just every now and then.
I've got another GP appointment to discuss in Feb, I'm going to ask about dietary changes such as excluding dairy etc, could you ask about that? I know it's not supposed to make a difference but I've definitely noticed a difference when she's had more milk, and I don't think she's lactose intolerant.
When it comes to taking him to classes, I would try something else like baby gymnastics or something else you haven't done to replace the swimming. Something where you can easily yank him out and change him if needed.
Sending solidarity to you- I know how stressful this is!!