But, Battlemaster abilities are far more consistent, and my first change is quite minor. The average difference between dice is 1, and this requires you to have rolled a 1 first, while at maximum value, and then you get a very minor bump in ability.
I'm not running the math, but it would be a decimal point increase in average power I think, nothing that would alter the balance enough to really matter.
It makes a huge cumulative difference.
Okay, I tweaked my simulation script so that it now calculates your average cumulative roll per day (basically, the average sum of all rolls), and so that it can optionally allow the die to increase by one size. That actually gets a better view on how much you get out of the d4/d6 split on rolls. As before, with the rules as per the UA you can expect 27 uses per day. Here's the stats:
As per the UA:
Attempting 27 uses per day, you can expect:
Median number of uses achieved: 27
Average number of uses: 21
Able to use the die on every attempt for the day 50.0009% of days.
Assuming all of those rolls are damage, you get: 72.4 extra damage per day.
Allowing the die to increase by one size:
Attempting 27 uses per day, you can expect:
Medium number of uses achieved: 27
Average number of uses: 25
Able to use the die on every attempt for the day 84.82% of days
Assuming all of those rolls are damage, you get: 100.7 extra damage per day
Remember, the expected damage per day from a battlemaster is 72. That puts the expected use as per the UA to be, on average, exactly the same as the battlemaster. With your update, you end up getting 28 extra damage - more than a third extra - over the day. That's a lot.