D&D 5E Impacts of allowing using a HD to reduce Exhaustion during a short rest break?

[Bolares]

I am thinking of adding a house rule which would allow the use Hit Dice to reduce exhaustion. Probably with the requirement of either using a use of a Healer's Kit or some form of healing magic.

In short, what problems might this cause? Beyond the Path of the Berserker's Frenzy, are there other class or archetype abilities this would notably impact?

It really depends on how prevalent exhaustion is on your games. In mine it's really rare for a player to have 2 levels of exhaustion (none of their classes have a feature that gives exhaustion), so that rule would make it basically irrelevant. if you table has too much exhaustion, and you feel it's getting in the way of the game, I think the house rule is viable.