The guy already wants to help you. The best thing you can do is be honest. "I think a lot of firms were concerned by my public interest work, but I am very interested in firm/corporate work." Tell him where you want to be, what you want to do, and make pleasant small talk.
He will likely forward your resume on to his contacts anyway. A lot of people are very well-versed in the job market favor-trade and do it all the time (even for people that they haven't met!). Whether it will have a lot of weight with the equity partner will depend on their relationship, how often he's pushed for people in the past, whether the people he's pushed for have been good or not, and whether or not he aggressively follows up (you're more likely to get at least an interview if he brings you up on a phone conversation vs. an e-mail forwarded to the secretary), the firm's position (if they're oversubscribed, they're oversubscribed). You can't control any of those factors; all you can do is have a pleasant lunch with this person and be friendly. You can also follow up with him, just tread carefully the line between assertive and pushy.