If you don't find a reliable university domain list, you might go with moderation: keep your own database of university domains, and only if you get an unrecognized domain, mark it as pending review, and have a real person verify if the domain belongs to a university or not, and then add it to your database as either belonging to a university or not. Then you could pre-populate your database with known universities (according to what you expect from your expected userbase), and blacklist major free email providers (i.e. mark as non-university), so you don't have to verify them one by one.
This approach has the additional benefit that you will end up with a filtered database of university-related email domains, which would be a useful resource which you can then offer as a service.