Think of hit points as "Amount of mistakes the player is allowed to make" and make sure that everything can be dodged and all damage avoided. So any thing that hits the player should be the player's own fault, sort of like Dark Souls.
If you go with a living character, you may want to really put some effort into the art and animation to reflect the character's weakened state. Limping, missing limbs, bandages wrapped around various body parts, etc. Scars are fine, but maybe more than just that. Then maybe some mechanical changes, such as taking longer to prepare a heavy attack, longer to recover from actions, longer mana regen time, longer cooldowns, slower movement speed (Thus harder to avoid attacks) to reflect the accumulated damage as well.
I wouldn't go with the bed thing though, it's very non intuitive that resting would make you weaker. A Rogue type game would be best for the mechanic...and possibly one where you're an android who "Woke up" in a post apocalyptic future. As a robot in a world without robots, it makes sense to the story why any damage winds up being permanent because nobody knows how to fix you. You then start with a rather large pool of armor/health and the goal is really to see how far you can get in your mission.
A compromise may be similar to what the Wolfenstein game (Xbox) did, where your health was 1-100, and any damage would heal up to the next 25. So if you were damaged to 80 health you were good and would eventually regen up to 100 again. But if your health dropped to 70, it would only regen to 75. Then you simply don't include healing items. This means that the player can afford to make more mistakes, while still maintaining the tension and aversion to damage. Opposed to Halo's "Energy Shield" system which similarly encourages players to take brief pauses from combat to regenerate, but offers no lasting penalty.
This of course means that you're not empowered and the story should reflect that. Again they need to either be able to avoid damage, or avoid the combat (Stealth mechanics).
I will offer a word of caution however, one of the big reasons why games moved away from "Health" to "Regeneration" mechanics was because the ladder are easier for the developer. As the designer, if I know the player will always be at full strength when they reach a room, I can design a good encounter based around that. If you do perma damage, then everybody who reaches that room has different health and abilities/weapons, so it's much harder to design an enjoyable encounter.
Yet another alternative (Lol) would be having it set up so that time doesn't pass in the dungeon, but the Only way for the player to heal is to sleep in bed, while limiting the number of in game days the player can play. The player still needs to avoid damage as much as possible, because in order to obtain the strongest possible end game state, they'll need to have longer grind sessions between resting. Again still keeps a decent stress level during combat encounters, but it's not brutally punishing. Days counting down will serve to increase the player's tension as the final showdown nears. So similar to many roguelikes, you need to make the most of the earlier levels to make it easier on the later ones.
ANOTHER (Okay last one, because I could seriously do this for a while) alternative would be say the player has 100 health. Any time they go "Critical" (Less than say 25% health) their max health is reduced. So they can still go to bed and heal up, and the long term ramifications are felt. As you're more and more scared, the tension increases to where you get to the point where even a single enemy attack will kill you, and then the tension just sky rockets. The victories feel amazing, and you know that defeat is just around the corner.
So yeah, you can absolutely make a fun game with limited health mechanics. Don't restrict yourself to what's been done before.
