This is the same approach I first tried, and it is the simplest setup. The selection works and it is saved in each computer. The problem that still persists is that B_PickRandomCharacter is owned by the controller, but it always loads the selection made by the host controller. I tried it again with multiple copies of B_Tinplate UE4 for each predetermined sphere each color, and it is not working:
The client always gets the same selection made by the host. That is the reason I believe the change should happen after possess event, called by the player character as shown above in this thread.