From the lore point of view, it would certainly make sense for longnecks like Littlefoot and Shorty to change their names when they grow older. Maybe it's some tradition with the longnecks to name their children with such nicknames for the durations of their childhoods. Of course, Littlefoot's name will never change in the series and I don't think it would be advisable to do in a fanfic but this childhood name theory sounds pretty sensible for me.
However, as Hypnobrai suggested, I don't think it is mandatory for a longneck to have a separate childhood name and if they do, I'd see them being able to hold onto it if they wish so (which Littlefoot might do). For example, there's little reason for Ali or Rhett to have a name change.