I assume this is possible because the rules state to play until the end of the round and short rests are included in end of round procedures.

I’d say it’s within the letter of the rules. And as we’re talking about 1 perk tick - why not?

You can change your lost card?? How did I miss that rule

Short rest: During the cleanup step of a round, a player can perform a short rest. This allows that player
to immediately shuffle his or her discard pile and randomly place one of the cards in the lost pile, then
return the rest of the discarded cards to his or her hand. If the player would like to instead keep the
card that was randomly lost, he or she can choose to suffer 1 damage and randomly lose a different
discarded card, but this can only be done once per rest.