Rather than giving monsters the average HP listed in their stat block, give them the maximum.
Now that I've solved the titular problems for you, what do I win?
You, sir, win 1xp.
But that only solves half the problem, that being increased difficulty.
How do you propose to solve the number of encounters expected per day?
Rather than giving monsters the average HP listed in their stat block, give them the maximum.
Now that I've solved the titular problems for you, what do I win?
3 rages per day. If you have 6 encounters, that means you can rage in half of them. Max. If you have 3 harder encounters, then you can rage in all of them.
One trick I stumbled into is having enemies run away, only to attack later. I forget the specifics, but the barbarian immediately raged in the initial encounter. Just a few rounds in, the enemies ran away and the barbarian was unable to pursue. They collected reinforcements and attacked the party again, only to run away again. By this point, the barbarian was really concerned, as he'd already burned two rages and the enemies hadn't been clearly defeated.
As with my idea in another post, I don't do this all the time. Just enough to keep players on their toes.

(Dungeons & Dragons)
Rulebook featuring "high magic" options, including a host of new spells.