Is it possible to have the Connect app use my custom HR zones when displaying time in each zone after completing a run? Right now it really doesn't seem to be working that way.
Check that your watch is using your custom HR zones before going for a run. Time in zones is calculated and recorded by the watch, and not evaluated by/in Garmin Connect after the data has been uploaded; changing the zone thresholds afterwards (either on the watch or in Garmin Connect) will not alter the record, and if the HR zones settings on your watch are incorrect on the watch while you're running, then the results will stay incorrect.