Automatically create scheduled reminders for recurring events?
Rather than create a scheduled reminder based on "Event Name", create it instead on "Event Type".  If you don't want to apply scheduled reminders to all events of this type, create a second type with a very similar name to use with events you want a scheduled reminder on - then attach scheduled reminders to only that event.

I can also see in the API explorer that there's a "cascade type", which seems to suggest that perhaps you could change the default cascade type to cause scheduled reminders to be cloned as well.  I'll let someone else speak to the feasibility of that.