Anamika,
That must be so confusing, both for you and the kids.
There is a lot written about memory and BPD. Managing intense emotions or hidden stressors on a near-continuous basis have a way of disrupting some of the cognitive processes, memories don't get stored properly, etc.
Either way, it seems like the problem is that she shifts responsibility to you, and that's frustrating.
How about when she mentions dinner again, email to say, "Remind me by noon the day you want to have dinner with them so I can get them there on time. If I don't hear from you, I'll assume you'd prefer a different day and we can reschedule."
Let her be the one to cross her Ts and dot her Is.
She beats me up constantly that she doesn't get to see the kids as much as she would like but cannot commit to a plan and be accountable.
Don't chase this stick if you can avoid it. It costs her nothing to blame you for what is ultimately her responsibility. No point in you paying the price.
Your kids see her falling down on the job, and that hurts and needs to be validated. But trying to shelter them from that won't protect them, it just teaches them to do cartwheels for difficult people.