I did, the calculation is pretty simple. Compared to a subclassless fighter the brainless-BM works out as +18 points of damage per short rest (levels 3-6), whereas the Champion (with 2H sword) works out at about +0.6 per hit.

It's still a fairly meaningless calculation - no player who is interested in tactical combat will want to play a champion anyway. It's like trying to balance the tree that falls in the forest.

Yes, that calculation was done.

The other one is the BM who manages to get a riptose strike for every die. So 4 extra attacks (and 4.5 damage per hit on them) per short rest.

We could also balance the half-orc champion for +1.2 damage per hit.

At 4 rounds/combat and 3 combats/rest that is 12 rounds for the champion for 7.6 damage per short rest over subclassless fighter, compared to 18 for the simplest BM.

Assuming 16-18 stat and 2d6 weapon and attacking 14 AC at level 3-5, each attack is +5or+6 for 2d6+3or4 (10/11). Each attack is worth .05*7 + .6or.65*10or11 = 6.35 or 7.5 damage. Extra dice on it are worth .65 or .7 times die size, or +2.925 or +3.15. So the BM riposte is worth 9.275 to 10.65 damage for a die.

The riposte-BM gets 4 attacks for 37.1 to 42.6.

At level 20, you have 6 (and maybe) more dice per short rest. You might be using a flaming greatsword for 4d6+5 damage. Enemy AC is going to be 18ish, while your attack is +11. So an extra attack is 4d6+5+1d12 or 25.5 damage and +20.5 on crit, for 70% accuracy and 5% crit or 18.875 damage times 6 is 113.25 damage per short rest (plus some extra if you activate Relentless, but that is dangerous).

The level 3-4 champion is getting 7.6 damage. The level 20 champion with a flametongue and some source of perma-advantage is getting .177 * 14 * (4 * (3*4 + 2)) = 138.768 damage, without .1 * 14 * (4 * (3*4 + 2)) = 78.4 damage. But I didn't include advantage in my calcuation of the riposte-BM.

Riposte-BM has 1.44x damage at level 20 and 4.88x extra damage at level 3. Again, the gap is much much larger at level 3 in scale terms. In absolute terms, it is actually pretty flat: 29.5 damage vs 34.85 at level 20.

In terms of swings, it is 4.6 swings/rest gap at level 3 and 2.4 swings gap at level 20.

A Flaming sword with advanage and GWM against an AC of 18 is .75 hit, .1 crit for 35.5 on a hit and +20.5 on a crit, so 28.675 per swing, a 1.52x scale on the extra damage on a BM. The Champion's extra damage scales by 1.77x in comparison. The gap falls to 33.37 damage in favor of the BM with perma-advantage and GWM, but this is only like one and a half swings.

Gaining an extra attack on an action surge gives more swings at 20 than at 3, so it leans the wrong way. Getting more swings on a second wind is the same number of swings per rest. Damage per hit gives more than 4x the damage at 20 than at 3, so also is a poor fit.

Damage on crit is like 6x larger at 20 than at 3, so that is also a bad place to put power (worse than damage per hit).

**Per-Hit damage**: 4.3x as much damage at 20 than 3 (gap mostly unchanged with advantage)

**Per-Crit damage**: 6.5x as much damage at 20 than 3 (gap larger with advantage)

**Extra Swings**: 2.3x as much damage at 20 than 3 (gap larger with advantage).

**Extra Swings on AS**: 4.5x as much damage at 20 than 3.

**Damage Gap needed to fill**: 30 at level 3, 35 at level 20.

We could aim for the gap to be on average closed at 3 and 20 -- be under at 3, and equally over at 20. To make this as small as possible, that is extra swings.

Our overall target is +32.5 damage. Swings are worth an average of (6.35 + 18.55)/2 = 12.45 (without features, using typical stats/weapons). 32.5/12.45 is 2.61 extra swings per short rest.

That is pretty close to 1 extra swing per encounter.

We want to keep this simple. And I want to avoid this scaling insanely with multiclassing. So how about:

**Action Hero**
Starting at 3rd level, a Champion can make a weapon attack as a bonus action on the first turn of combat. It they qualify for the Two Weapon Fighting bonus action attack, they may instead make up to two weapon attacks as a bonus action, but no more than 1 with each weapon.

I think that is easy to remember and doesn't require tracking resources, so should be suitable for a Champion.