I would think that placing the pushpin by street address would get around your problem... However, it does require additional information and, therefore, extra effort. And, that information may need to be held in confidence. You could simply use false addresses on a main road in a given zip code...
For data calculations, you might want to export the pushpin data set to Excel. Then, sort by zip code and count by zip code...
You could import from existing address spreadsheets and have Streets & Trips set the pushpins for you... I am not an expert on these procedures... But, others on this site are. They are VERY gracious and would likely give you pointers, if this sounds like it might work for you.