Your problem isn't hit points, it's how much power is gained through leveling. I suggest starting a game at level 5 and having level 10 be the max achievable level. Characters level much more slowly, their hp won't ever double. Their damage ouput won't double. Modify XP gain if you want a long campaign or a shorter one. I think that solves almost all of your issues.The mechanic that most bothers me with D&D ingeneral is the use of hit points. yes, this is well trod ground for internet debates, but I am interested in actually finding a solution to the "hit point problem."
Now, it is not hit points per se that bothers me, but how they increase significantly as the PCs gain levels. It is primarily through hit points that characters become superheroes in D&D, able to shrug off attacks that very recently would have been worrisome if not downright fatal. And note that this isn't about realism or trying to simulate anything. I simply don't like how the change happens, and the rate at which it happens: the PCs head off to the Sunless Citadel and have to carefully pick their way through the wilderness lest they run afoul of bandits, for example, but three days later they are 3rd level and the way back is barely a concern because those CR 1/8 bandits are no longer much of a threat. That change in threat is mostly about damage capacity.
One solution is to effectively star PCs off at more hit points and then greatly reduce the number they increase over time. This means threats early remain threats later and PCs increase primarily in versatility and skill rather than toughness. But figuring out where to put that starting value is highly dependent on where you expect the campaign to end. If you are running a 1-5 like Phandelver or DragonHeist, you can give everyone max 3rd level hit points and then allow them just their con modifier per level and you should be good. But if the end is in the teens, balance is going to get really wonky at low levels.
Another solution is to use something like the Mutants and Masterminds damage save, which seems to work well enough for super hero d20 games, but might require a lot more initial design changes to make balanced and workable.
Have you eliminated hit point inflation from your 5E game? How did you do it? Did it work?
Alternatively only allow PC's to level during extended downtime and don't tie adventuring to leveling. Tie adventuring to loot.