From your suggestions, the best answer is: "Je retrouverai mon ami".

"Je me retrouverai avec mon ami" means "I will find myself with my friend".

"Je me retrouverai mon ami" does not mean anything.

It could also be translated as "Je vais retrouver mon ami".

Another suggestion might be "j’ai rendez-vous avec mon ami". You can say "se retrouver" when you mean both yourself and the other person, as in "On se retrouve dans une heure (à tel endroit)". Otherwise, it’s not reflexive, you just say "je retrouve un ami", "je vais retrouver un ami".

The only true grammatical error lies in dropping "avec" after "se retrouver".

Now, in daily life, the most typical informal way of translating "meet up" is not "retrouver" nor "rendez-vous" (they meet a friend!), but "se voir".

"On se voit au magasin" Let’s meet at the store (when concluding a conversation)

"Je dois voir Cathy à 14h" I will meet Cathy at 2pm, I have planned to see/meet with Cathy at 2

"Quand est-ce que tu as prévu de voir Mickaël?" When is it you are supposed to meet with Mickaël [again]?"

"Je vois mon amie tout à l’heure, tu veux que je lui passe ton bonjour?" I will meet up with me friend in a bit/layer, do want me to say hello for you?