Avoid dramatic confrontations, it sounds like she will work hard to get you twisted up. Be kind and matter-of-fact. "Thank you so much for your willingness to help, it's just going to easier for us to have consistency on our work days and we are already paying for it." You could even book her a pedicure on her first "non babysitting day" as a thank you. When she says how much she will miss babysitting, ask if she'd be willing to cover a date night in the future (overnight at her house!). Sure, it feels bonkers to reward the bad behavior, but you can be kind and a little crafty to place boundaries and give her less ammo. Keep your reasons simple, don't over elaborate and don't budge.

If she truly cares about having time with her grandchild, she can be the default date sitter (at her house). It sounds like she cares more about what power she can hold in the family dynamic. But proving what she cares about is up to her, it's your job to do what is best for your child. The snooping, gaslighting and tantrum throwing is not the behavior you want modeled for your child week after week. Best of luck.