Hello Caroline. What is the price range the customer had in mind? Have you talked with them about maintenance and what they are looking for there? Do you think a sponsor unit may not require 20% down?
Here's the thing, building management companies that may have unsold shares are less likely to allow exclusive listings as they feel the more brokers they can get to bring over prospective customers, the better. With that in mind, most brokers/agents who are aware of such coops are not usually interested to give away something they (possibly) can't co-broke and get themselves some commission. Your best bet is to pound the street and take down information as to who manages a particular building, then call them and ask. Be aware that some (many) sponsor units may need updating.
Lastly (and I don't mean to be flip), if you truly have a "qualified" customer, why are they concerned about a board interview? Interviews are mere formality as any board would have a board package with full financial disclosure of the customer prior to any interview. If you get an interview date, you're pretty much home free. Good luck.