When you say a 'show name', are you speaking of a registration name or a name to show at local shows where registration isn't necessary?
Obviously the chestnut horse would not be able to be shown for a long time, due to his condition. Up-to-date conformation pics would help when thinking of a registration/show name.
Do you like the pinto's current name or wish to include it in a show name?
Do you have a farm name? If so, you could name him something like, Green Valley Farm's DayKotah. Not sure I like DayKotah. If he were mine, I'd call him Harlequin Farms Best Dressed or Harlequin Farms Winchester. For a call name, you could call him anything which is easy.