Interviewing in MM/UMM PE with Non-Traditional Background

I am currently at a MM bank that has decent exits to PE (usually LMM, some MM) and I am shooting for squarely MM PE and based on my conversations with recruiters, I think its doable. I come from an unconventional route to banking (non-target undergrad, MSF) and was seeking advice on how to position myself. 

I had mostly written off myself for any top MM or definitely UMM funds but if I was to somehow convince a recruiter to move me into the process at one of these firms, would they take me seriously if I really killed the initial interviews? I have grinded to feel very confident in my technical interviewing skills and I think I have always done well behavioral-wise. I am also a top performer (I am a 2nd year, ignore title) and have some solid deal and modeling experience relative to others in my group. I want to go for the best opportunity possible without burning my reps with firms that wouldn't truly consider me due to background.

Any other general advice for positioning myself during the PE recruiting process without the ideal background would be much appreciated.