But that law contradicts what you say ("If the keeper is injured, one of the outfield players must stand in for the shootout."). It says 'With the exception of a substitute for an injured goalkeeper,' clearly saying that, yes, a keeper can be substituted during a shoot out.
A goalkeeper who is injured during a shoot out can be substituted only if the side haven't used all their substitutes.
https://www.fifa.com/mm/document/afdeveloping/refereeing/law_14_the_penalty_kick_en_47369.pdf