The hink-pink for "The extra cash that Santa keeps in his shoe is his" is "Yule mule." In a hink-pink, the goal is to create a pair of rhyming words or phrases that answer a riddle or description. In this case, "Yule" rhymes with "shoe" and "mule" rhymes with "cash," making it a fitting hink-pink for the given statement.
boot loot
cash stash, boot loot
boot loot
The hink-pink for a 'a brief rest for Santa' is a Claus-Pause.
One hink pink for Santa's cut facial hair would be Santa's sheared beard.
Kringle's Pringles
Claus floss
sled bed
Claus Floss
quick St. Nick
Claus floss
spare chair