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. [...]
Each of the three following Blade Flourish options say:
You can expend one use of your Bardic Inspiration 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.
Is the damage from the Swords bard's Blade Flourish feature doubled on a crit?
Each option also has a unique secondary effect based on the number you roll on the Bardic Inspiration die, but I'm specifically interested in whether you roll twice the number of damage dice on a critical hit.
I've been having a small debate with another player, and they have a small belief that it is not. I was doing my own research and everything I saw pointed out that it would be doubled on a crit, because they are rolled damage dice. However, the other player argues that it's additional damage - that you are declaring it and rolling the Bardic Inspiration die for it prior to confirming whether you hit or not.
Based on my own research, this issue has been rarely touched upon. Even on Sage Advice and the EN World forums, 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 as no different from a Battle Master fighter's superiority die roll, or a Paladin's Divine Smite. Both are effects you declare you are using after you have attacked the enemy, and are thus subject to the critical hit rule, which doubles the number of damage dice.
The counterargument is that the die (Blade Flourish) is not tied to the damage roll; it's an addition that's applied after you figure the attack's ability to land.