Solution: Collect data on every action that anybody does during a trial, Bucket finished trialers by "Converted" and "Expired". Visualize that data to find out what motivates trialers to convert. Run crazy ML magic on it to predict outcomes for individual trialers. Act on that information.
About half a day into building, it occurred to me that this is not in any way specific to the one product I was building it for, so I moved the API endpoint out to its own domain. There's a general purpose product there in beta now: