Hi.
Regardless of where you meet or who pays, I would not expect an apology. If you get one, great. If not, then accept that you did not and probably never will. Some people choose not to apologize. Some people can't apologize.
I don't want her to pay for the meal and then use it against me but I'm afraid she'll get mad if I insist on paying my bill. I don't want to give her anything else to use against me. It's very hard for me to stand up for myself in person to my mom because she triggers me and upsets me. I wish I hadn't agreed to see her but I'm close to my dad and really want to celebrate his birthday with him.
I understand not wanting her to get angry. I also understand how scary and triggering it can be to say no and stand up for yourself.
Why not use this as an opportunity to change things up and practice asserting yourself? One possibility is to say "okay, you pay, I will leave the tip". Don't ask, make a statement and do it. Another option would to say to the waiter that you want a separate check when they take the order. She may say something. She may get angry. Let her. Her reactions do not, or should not direct your behavior on this.
I think this meeting is a good place to practice standing up for yourself. Setting a limit. It is in public. There is a built in time limit for contact. You will have your own transportation away. You will have your own money.
In other words, you will have a whole lot of options open to you and with options there is power. You do have power here. You are not helpless. You will survive her anger (if she gets angry). If you get triggered you will survive that too. You have us to help you get through it. No contact is good for giving you space to heal for a limited time but it is not necessary to heal.
Do you want to practice things you can say during the meeting? What sort of scenarios can you anticipate?