As far as I know, the event-spirit data (the event-spirits' images and names and the triggers, which tell the game, whether or not the event-spirits can show up e.g. on the spirit board) is stored in the BCAT save data. When you backed up your savegame from your SysNAND - did you back up the BCAT save data as well? JKSV lets you do that. You can simply restore the complete BCAT save data on the EmuNAND again using JKSV. Afterwards the event-spirits should show up with names and images.
(Source: I did that myself, so my EmuNAND save is complete with all event-spirits obtainable)
