Class I seems to be the biggest problem. Executioner has 8 tons of JJ's, yet has trouble jumping up on platforms on the mining collective.
To fix JJ's I'd suggest a speed based jumping mechanic as told by gifs:
1. If a mech is at a running speed and hits JJ's, 0-10% vertical bonus--hoverjets remain pretty much as is

Didn't jump high, jumped long
2. If a mech is at a walking speed and hits JJ's 50% of vertical jump bonus

Jumped higher, not as long
3. If a mech is stopped and hits JJ's, 100% of vertical jump bonus

Jump high, not long
4. If a mech is falling, no vertical bonus

Silly locust, you cannot fly
The jump bonus would need a slight delay and crouch from the mech as if it were actually leaping (yes, mechs can jump without jumpjets, they just don't stay off the ground very long). The vertical bonus would come as a bonus to vertical acceleration, instantly accelerating the mech to 3-5 m/s in the vertical direction (value to be tuned for balance).
I am for either reticule shake or lagging control response while falling, as the mech needs to remain balanced (see last gif for dude flailing arms to stay upright).