So, how about using the BHR as a damage roll for certain things…
Maybe lack of food and water should cause you to lose BHR per day without water, and BHR per week without food (distributed 1/7 per day).
Other places it doesn’t make sense for a 10th level character to be able to shrug off the effect, too, maybe, like falling damage, maybe?
I think that's a great idea. Any time it's a situation where heroic fighting skill wouldn't make a difference, BHR could apply.
E.g. Damage equal to 1/2 BHR per 10' fallen; damage equal to 1/2 BHR per day without food and drink; deadly poison could be 2 x BHR; and so on.
Makes sense. Base ACKS had kinda already made the argument that HP is a pool of "resolve to fight"; perhaps, and not actual health, as stated with the Mortal Wounds table.
So anything that reduces that resolve - starvation/dehydration, "system shock" sorts of things, the wind knocked out of you by a fall, etc. LOTRO expressed its HP pool as "Morale Points", and drained them both by hits and effects - canonical example being in the presence of a Nazgul - and restored them via rest or inspiration, the game has a "Minstrel" class as its healer.
Not that I want to encourage bards.
Instead of assuming you're encouraging bards, assume you're encouraging the best thing (I know that sets the bar low for most but hear me out) to come out of 4th edition: The Warlord!