I know you could be tempted to say that the player must be sent-off, but Law 12 says the following (Law 12.3):
A player, substitute or substituted player who commits any of the following offences is sent off:(...) receiving a second caution in the same match
The point is, kicks from the penalty mark are not part of the match. This was written quite clearly in the old Laws of the Game (from the 2012 LoTG):
The kicks from the penalty mark are not part of the match
But this is actually still true even if the same passage is worded differently in the current Laws (Law 10.3):
Kicks from the penalty mark are taken after the match has ended
Therefore, according to the rules, the player cannot be sent off because he has not received more than one yellow card during the same match. Does it mean the referee will only be able to show the player a yellow card and report the misconduct to the appropriate authorities, but without being able to actually send him off?