This can’t be solved without candid communication. (And it might still not be solved if he happens to be narcissistic or a golf addict, but communication is still your best bet.)
Depending on how strong the inertia of this imbalance is, and how much potential he has to take the initiative with child care, this could take a while. Tell him that you need more alone time to recharge, and that you’ve noticed he reacts better when such outings are planned in advance, so you’re going to draw up a schedule. Then get concrete. Whether it’s an hour-for-an-hour trade-off or a golf-for-shopping swap, the point is to make him more aware of just how much more time he’s getting and, more importantly, how much he takes you for granted.
Lashing out at your sister will only give her more material for her drama. (Now showing: “I’m the Victim!”)
Suggesting counseling to her was probably a great idea in theory, but she obviously heard it as a judgment that she’s flawed and as a refusal to collaborate with her. The word “ridiculous” probably didn’t help matters, if you used it. I must ask, what’s so wrong with the hash-out plan? Giving in to a little listening – with some boundaries that you set – might buy you some peace if she gets the chance to feel heard.