mean relative

Restaurants are quite happy for individuals to pay separately to settle a group bill. My friends and family do this frequently. No need to work out how much we owe either, we tell the waiter what we have eaten/drunk and he works it out. Then we leave a group tip at the table.
Your SIL's behaviour will change once she realises you are not a push over, Catlover. Speaking up for the first time is the hard part, subsequently it becomes easier.

Is being "a bit highly strung" just another way of saying she will spit the dummy and have a toddler tantrum if she doesn't get her own way? She has had a lifetime to perfect her technique and everyone in the family seems to be avoiding the obvious - she's one of life's takers and an ungracious guest. Does she have any redeeeming features? Is she good company, funny or just a self centred bore?

You know what she is like so the basic questions are -
Why do you invite her to stay?
Why do you feel the need to be so fulsome in the gifts and what you pay for when you visit her?
Why is it difficult for you set limits on what you do for her and what you pay for when she visits you?

Lots of good suggestions from Grans. Only you can answer the questions but a guest who leaves you fuming and feeling so aggrieved is not good for anyone. The old adage about guests and fish stinking after three days comes to mind.

I stopped going out regularly with a group of friends, all women, because one always took advantage and it annoyed me so much.
The others muttered but condoned it.
She always chose the most expensive food anf bevause she would only drink fizz would have a bottle of cava, the fizz of choice in those days, to herself.
The final straw came for me when one night at the end of the meal she got up and went to the bar, came back, sat down without saying anything and when the coffee was served the waiter said, who is the brandy for?
That for me was it, I never went again and the regular meals out stopped soon afterwards, when the others decided enough was enough.
Her husband treated her like a queen and she expected from everyone else.

I'm still puzzling the "keep modelling good behaviour" comment??? Op is dealing with a full grown adult not a child an adult who is taking the pee, is selfish and entitled. My comment is don't invite her again and if you do have to visit her go empty handed and only pay for your own food/drinks if you have to go out. Cheapskate relatives pfft

I wouldn't invite her again. Sounds very rude to me.

An otherwise good friend always took advantage of the 'split the bill' rule because, as a non-driver, he always had a lot to drink and one of us always ran him home. He only ever brought a credit card and often took all the contributions, including tips, and paid only the bill on his card! Last time we all agreed to pay only our portions of the bill, with equal tip each, and insisted the tip was added to the receipt. Amazingly this time he managed to find cash. He also had to get the bus home. A bit hard, but after years of this behaviour we'd had enough. He's still talking to us, but we'll see what happens next time. I think you need to find a way of telling your SiL enough is enough!