here's the REAL trick!
spawn your pet,go to the cave to that rock, and re-log.
if your pet spawns with you, then re-log again.
you do NOT have to spawn the pet on the rock and get the DC message!!!
you only need to log.in when your char stands at the rock.
as soon as you re-log and don't see your pet (but it is blinking in your inventory) then you may exchange it to the lower character.
you should maybe give more pets to your low char since lv +/- 80 isn't that much