I think that fighters that have superiority dice should get the options to add them to non combat ability checks.
Some kind of Grit or heroic effort mechanic, like proficiency times long rest fighters can increase their jump or give them selves advantage on Strength checks.
Mid to high level fighters should be able to gain some inspiring presence or leadership ability to aid the party resist rear effects or intimidation checks.
tbh the psi knight is what I would base it on... but I would give some things instead of spend a die "as long as you have 1 or more die"
give them parry reaction (self or adjacent allie) reduce damage by die roll+dex mod... special if you have a shield roll 2 dice. but this doesn't cost a die you just have to have 1 left.
give them the ability to as an action when they have to make a str/dex/con check (or skill related) they can roll the die and add it to the d20 roll. but this doesn't cost a die you just have to have 1 left
give them might leap. spend a die, roll it add 5x the die roll to feet you jump
give them super speed. as long as you have a die you can add it to initiative as long as you are not surprised. You can also spend one and add 5x the roll to your ground movement for 1 minute (concentration)
give them extra damage. Once per turn when you hit with a weapon attack spend a die to add it to your damage.
give them a spider climb like ability. As an action spend a die, roll it and for up to 10 minutes (concetration) you can move up walls and along ceilings with no problem (just being strong grip str) at a speed of die rollx5ft
give them mage armor (well initral armor form the 3.0 psionic book) as long as you have a die you have your base AC calculated as 13+dex (alternatively you could just give unarmed defense of a second stat) helping unarmerd fighters
Here is a BIG one... give them Legend Lore... yup a 6th level divination spell. Spend a die and 10 minutes investigating and thinking... it brings to mind a story, a parable, a hint a clue.
lesser then that but even detect magic/identify. As long as you have a die you can as an action feel for the flow of magic, get detect magic at range of touch... spend a die to recognize the feeling an item gives you to identify it. (tbf I think detect magic should be a cantrip anyway for casters too)
I can't belive it's not lightning bolt... Spend a die roll it adding 5x the roll to your movement, then move (as part of this action like a dash) and every creature you end up adjacent to during this move you can make an attack against (1 attack per even if you pass the same target twice) and if you hit or miss add the number rolled (die for speed) to the damage (so yes damage on a miss)
mountain hammer. Spend a die, then make an attack (so if attack misses you still lose die) if the attack hits add a number of d6 equal to your prof to damage, and all of the damage ignores resistance.