Emailing Partner at top EB

Hello fellow monkeys,

This might seem like an odd post but I would like to ask for your opinion on something and would be very grateful if one or some of you would take the time to give me their opinion.

Im an undergrad at a target school in the UK applying for spring internships with the objective in mind to convert (particularly interested in rx as I want to later move to turnaround pe).

I was lucky enough to meet a partner at an EB (think CTV, EVR, LAZ, PJT...) at a rather exclusive networking event a month ago. I had quite a long one on one discussion with the guy who was very kind and gave me lots of good tips (also very chilled/laid back guy so that was nice as well). I followed up with a short email to which he replied that "it was good to meet me too - best of luck with your application".

I am now applying to the firm for my internship and whilst I haven't yet heard back I was wondering if it would be appropriate to message him to either ask for coffee and then for a reference or directly for a reference. By reference I mean him boosting my application internally. The guy is very busy im sure so asking for coffee might not be the best.

What are your opinions on this? What should be my course of action?

Thanks so much !!!