SIL's schedule is set three weeks in advance. With adequate notice she can arrange the occasional Saturday off, liberating the ILs. She probably works Saturday because she gets a premium for it. She'd probably work Sunday, except OP said most businesses are closed Sunday where they live. So how much notice do SIL and ILs need? Right! More than three weeks.
If Saturday with Nephew is OK with the grandparents, they are likely giving SIL a boost and want to do this. Most people would accommodate this commitment most of the time.
OP lives an hour away, and wants to figure out, basically, how to schedule with ILs because, apparently, OP wants alone time with the ILs and SILs weekend schedule with the ILs makes that impossible, in OP's opinion. They have scheduled the playpen date, so this is now for future use.
Nephew is a brat. Nobody likes him or wants to be around him or take him anywhere for this reason. This is an etiquette problem. He needs to learn some. He isn't going to learn better until someone teaches him. As people who see him rarely, OP and DH are in an excellent position to give Nephew alternatives, to show him better ways to be heard and to get what he wants, because they are now outside the established dynamic. He is also a Big Boy now and Big Boys know these things.
"That doesn't work here", or "work with me" is usually enormously effective when followed by ignoring. "I told you before, that doesn't work here. Figure out a way that works" is the followup. "Nope, that doesn't work either. Try again." By this time, IME, the tantrum is stopped because the child is into the puzzle. Letting the child know you know the bad behavior is a tactic to obtain a reward is key, IME. When they demonstrate the correct behavior, which I tell them once they calm down, if they have not guessed, they get the reward. Six is not too young to learn the better way, the polite way, the one that works with you, or at your house, or in society, quickly. I have stopped tantrums on the part of habitual throwers (who were in therapy for this) in their tracks with this very dispassionate and business-like counter strategy.
I do not like tantrums and they do not work with me. Letting children discover, or even telling them straight out, what tactics will solve their problems and get their needs met in the real world is effective because that is the real issue: they simply did not know what worked (was polite) because no one had never told them. Teaching Nephew manners will change the whole dynamic and might just solve the problem. It is a little work, but it is also a public service and good practice.
If SIL is a pill, that exact form of behavior modification works on bratty adults, too. OP could also make it Saturday visits with the ILs to avoid her. The key to solving this seems to be teaching Nephew some manners and the adults acting adult. The underlying issue of sibling rivalry is best worked out by DH in therapy.