(2014) Can a Battlemaster use Piercer's reroll with a Riposte attack?

The Battlemaster Fighter has a maneuver called Riposte.

Riposte. When a creature misses you with a melee attack, you can use your reaction and expend one superiority die to make a melee weapon attack against the creature. If you hit, you add the superiority die to the attack’s damage roll.

Piercer is a feat from Tasha's Cauldron of Everything, and it includes in its text:

Once per turn, when you hit a creature with an attack that deals piercing damage, you can reroll one of the attack’s damage dice, and you must use the new roll.

Piercer says you can only reroll a damage die once per turn. There's an interesting interaction with Riposte, which is an attack you typically make on an enemy's turn, when they miss you. If you use the Riposte maneuver (with a weapon that does piercing damage), does Piercer apply, allowing you to reroll one of the attack's damage dice? Can you only do it if you didn't use Piercer on your previous turn (or don't use it on your next turn)? I'm mostly concerned with this in a 5e/2014 rules campaign, but if there's a difference between 5e and 5.5e, that'd be interesting to know as well.

Quote from Stabbey_TC >>

If you use the Riposte maneuver, does Piercer apply, allowing you to reroll one of the attack's damage dice?

Sure, as long as the Riposte attack is dealing piercing damage. Nothing says you have to do it on your turn.

Quote from Stabbey_TC >>

Can you only do it if you didn't use Piercer on your previous turn (or don't use it on your next turn)?.

There is nothing in either feature that says anything like that, so there is no such limitation. "Once per turn" means "once per turn"; it doesn't mean "once per your turn". In this way it works like the Rogue class's Sneak Attack feature, which is also "once per turn" but can be done on anyone's turn.

Quote from Stabbey_TC >>

I'm mostly concerned with this in a 5e/2014 rules campaign, but if there's a difference between 5e and 5.5e, that'd be interesting to know as well.

Regarding this specific issue, there is no difference between 5e and 5.5e.