I've considered similar - to the point that I reworked exhaustion.

I think, if you were to adopt this, I would suggest keeping the exhaustion levels the same, but rename it. You're looking for the mechanical effect of a Wounds/Vitality system, where you've run out of most of your hit points, and are down to "wounds." Which is fine, but calling the effect exhaustion just leads to confusion, and overexplaining what you mean.

Creating a new name for the temporary effect would alleviate most of the confusion.

I'd suggest something like 'On death's door', or 'Battered'... Could even use the old 'Bloodied' nomenclature for being at or below half hit points to let people know you're getting close.