“The question is does a positive commandment override a negative commandment in this case?”
You’re never allowed to. At least not in the way that you’re talking about. It would be a “Mitzvah haba al yedei aveirah” and it’s assur, like using a stolen Lulav.
In any case, the concept doesn’t even apply here. Dating is not a Mitzvah in and of itself. It’s hishtadlus. It would be like working on Shabbos in order to earn a parnassah.
Hishtadlus is not allowed to be done by means of an aveirah.
In terms of your first paragraph, the implication is that one is allowed to deliberately commit an aveirah for the purpose of finding one’s zivug. That has nothing to do with “shiluchei Mitzvah ainan nizokin”.