“auprès de” really means “with you”

“à côté de” is more like, standing next to you, or sitting in the train, it’s less romantic! It does not really make sense in this sentence in french

Moi je ne dirais ni l’un ni l’autre, mais :

Tu es celui / celle avec qui j’ai envie de passer le restant de mes jours.

Ceci dit, auprès de laquelle serait le bon choix dans le contexte amoureux. Et dans aucun des cas il ne peut y avoir d’ambiguïté sur le sens de rater.

In terms of grammar, both sentences are correct, and so is the sentence suggested by Laure.

However, as someone mentioned, there is an expression, written "aux côtés de", that would fit perfectly in this context. This expression is the figurative version of "à côté de". It means "to be at someone’s side, but not litterally". In your example, you don’t need to be next to your beloved one all the time to be considered as "à ses côtés". This very subtle difference is discussed here.

In my opinion, "aux côtés de" implies a form of permanent support that you give to the other person. It means that whatever happens, you are standing next to him/her, no matter if you are with him/her physically or not. For this reason, this is the expression I would choose in your very context.

Tu es la personne aux côtés de laquelle je veux passer le restant de mes jours.