Your mother seems to want to throw a BBQ that has nothing to do with your birthday. I'd tell her that's fine because of course she has that right, and it doesn't seem like you asked for a celebration anyway, but "Please do it when I'm not in town so I don't have to spend time with the Awful Woman you are determined to invite. Let's have my visit be pleasant, please. Or, if you absolutely need to have the BBQ with Awful Woman while I am visiting, I can find somewhere else to be during that time." (That's if you really agree with that last part. You might just want to visit some other time.)
I feel very strongly about people trying to force people to spend time with people were cruel or abusive toward them, especially when it's an adult who mistreated a child (no matter how old the mistreated person is now). And, while I understand the urge as the mistreated person to try and avoid saying the truth out of "politeness," because I have done that my whole life up to this point, this is something I am now trying to change because it only protects the Awful Person and places part or all of the blame on the person they mistreated.
So, if anyone asks about your or Awful Woman's absence, I would ask your mother to not say anything "diplomatic" like "Awful Woman and GSNW just don't get along anymore," because that's not the reality. It's not a "two way street" and has nothing to do with any decision you made in your treatment of her. This isn't someone who accidentally tripped and spilled something on you, or disagreed with you about what TV sitcom was best, and you stormed off in a huff and unreasonably declared you no longer wanted to have anything to do with her. This is someone who purposefully hurt you, and you should not be held in any way responsible for that.
If anyone is crass enough to demand an explanation for why Awful Woman was not invited to your birthday party, or is wondering why you are absent from the BBQ while you are in town, I think your mother should tell them the truth: "Awful Woman does not like GSNW, so I did not invite her to the birthday party. Beandip?" or "Awful Woman does not like GSNW, and has shown so in her mistreatment of her, but it was more important to me to have Awful Woman at my BBQ than GSNW. Beandip?" (This is if you're comfortable with that.)