Ex-Friends

So this girl (Brooke) and I were best friends since 6th grade. She actually came and lived with me the summer after 8th grade bc of some personal things her and her family were going through. When she went back home (9th grade has started) a few months later "she" (the whole thing was actually her mom who can't stand me bc I didn't shelter Brooke from reality like telling her what condoms were not like condoning her to have sex, I'm a good kid) started sending messages to these people we had class with and saying my 60yr old uncle supposedly was sending her nudes (he has a government phone with no camera) and a bunch of other bullshit. Long story short , about a year ago me and Brooke talked over Instagram and we made up to be acquaintances. Well now we have awkward eye contact in the halls at school and her and her boyfriend came to sit with my friend and I at lunch a few time and everyone is talking except her. I really want to try being friends again with her, bc before all that we were inseparable. Idk what to say to her though, my other friend is like just go up to her be like hey wanna hang out? But I feel like it's not that easy, but I also feel like I'm making too much of a deal out of this... I just don't know what she's gonna say or if she'll be happy or not. Any advice on what to say or anything helpful? I plan on going up to her at lunch which is in an hour.