I’d suggest changing the field that holds your class from a Multiple Select into a Link to another record field. This will require a new table, “Classes”, that has a record for each class you are offering.
Your registrants can still see a list of all the classes available in a very similar manner to the multiple select list they had before - they can still choose multiple options. The difference is, from your “Classes” table, you will be able to build a roster of all the names that are registered for each class, since each class is a separate record.
Try it out and ping back if you need further help.