I have not done too much thought if this is a good adjustment or not, but what if:

• +No longer an exhaustion perk
• +Increase the window you can take a hit (0.5s -> 1s)
• +Can use it when either healthy or injured
• +/-still has a 40 second cooldown after using it, if you have not taken damage during DH
• -It's a one time use. Taking an endurance hit will disable the perk for the rest of the game.
• -Perk is disabled in EGC

This would design the perk to give the survivor one more healthstate like current DH. However, the change is that it's only one healthstate. not multiple hits you can take over the course of the trial. Perks that gives Healthstates are always powerful, and the ability to potentially gain one every 40 seconds is too powerful for a single perk.

DH was previously adjusted to make the window you can use it smaller to "nerf" it, but it only just made it so it has a more difficult barrier of entry while not affecting the higher skill bracket of players. Reverting it to be easier to use should be ok if the perk is single use. This would put the perk closer to DS or adrenaline, so all 'free healthstate' perks only gives one healthstate only. (DS should also revert the 3s stun back to 5, but that's another issue.)