Having a small debate with another player and there is a small belief that it does not. I was doing my own research and everything I saw pointed out that it does in fact double in dice, and therefor should be treated as a crit since it’s rolled dice. However the other player argues that it’s additional damage, that you are declaring it and rolling the bardic inspiration for it prior to confirming whether you hit or not.
In my own research this has been rarely touched upon, and even on Sage Advice and enworld the most I’ve seen is people saying “the damage can be doubled but not the effect”, but it was only a couple of posts.
I’ve always seen it no different from a battle-masters maneuver roll, or a paladin’s smite, both being effects you declare you are doing after you have attacked the enemy, and thus subject to the critical clause of being damage doubled.
The counter argument is that the die (Blade Flourish) is not tied to the damage roll, its an addition that’s applied after you figure the attack’s ability to land.
Blade Flourish (Edited for brevity)
Whenever you take the Attack action on Your Turn … if a weapon Attack that you make as part of this action hits a creature, you can use one of the following Blade Flourish options of your choice.
Defensive Flourish. … deal extra damage to the target you hit. The damage equals the number you roll on the Bardic Inspiration die. You also add the number rolled to your AC until the start of your next turn.
Slashing Flourish… deal extra damage to the target you hit and to any other creature of your choice that you can see within 5 feet of you. The damage equals the number you roll on the Bardic Inspiration die.
Mobile Flourish. to cause the weapon to deal extra damage to the target you hit. The damage equals the number you roll on the Bardic Inspiration die. You can also push the target up to 5 feet away from you, plus a number of feet equal to the number you roll on that die.