If you want to disincentivize the whack-a-mole effect a common houserule is +1 exhaustion when you hit 0.
If you want an actual Death Spiral a possibility is 1 Exhaustion when you drop to 75% HP, 2 at 50% and 3 at 25%. Just make sure that this doesn't stack with spells and other effects that influence Exhaustion. It's a fair chunk of extra bookwork though, be warned.
Edit: Another possible death spiral is manipulating prof bonus as you lose HP. For example Half Prof after you're down 1/3 your HP, no prof at 25% HP or less.