Morning Matthew
I've never had any issues when changing a course name, it should also update learners record of learning to the new course title as well.
I always update the short name to the same new course title as well as I would think that may be confusing as the short name would be shown with the old title in some places i.e. the calendar entry if you use the Upcoming Events block.
You may want to check if the old course name is used for any search filters for reports/scheduled reports.
Regards
Sue