I'm guessing to SIL & possibly MIL, the poor little boy is an only child and you give him the opportunity of socialising, BUT, he gets this chance of socialising with other children with minimal effort from SIL.
It's not your responsibility, although as a loving aunt you should embrace him with open arms, as what's another child when you've got three already right? Crazy!
I would say to MIL you love the children socialising, he's the most lovely little boy, why doesn't SIL come and get your children and take them out. Maybe then she'll realise that he's not just another small person hanging around, you have to make concessions for each and everyone. If you're going to an adventure playground you still have to pay for his entrance fee.
If SIL is overwhelmed with 1 child I would suggest she considers no more. Wouldn't it be nice for MIL to give you a call and ask you to drop off your DC, so you can have a bit of time to do whatever?
I think the only way to solve this would be moving, which sounds extreme, especially since you get on with MIL, plus DN isn't that bad. SIL should really consider common courtesy, even if texting, popping round to Mums later, would you mind if DN played with your DC?
In a way it's not as much SIL that I'd worry about upsetting, as you gather your DC and rush out to avoid the inevitable. She may have been (unfairly) saying oh you get to play with your cousins later.
Part of me thinks that seeing as he isn't badly behaved, also that during school holidays, unless he has local friends he's going to mostly be alone, interaction with his cousins is special time.
I would say if you're planning on going somewhere, warn SIL and say I'm sorry, we're fine taking him, but can't afford to foot to pay for DN.
I think also cousins have a special bond, if he doesn't get anymore siblings, they'll be the closest he has to siblings.
So I disagree with SIL behaviour, but I think it's something nice for DN to share time with his cousins. Maybe reiterate, you have no problem him being there, just a bit of warning would be nice. If you sprung on her doorstep with your 3 DC I doubt she'd be as enthusiastic.