Yes. The breath weapon is in relation to the dragon. The dragon is still getting rounds to do "things" in. Likewise, if it had cast any short duration buffs, you would tick off rounds of those buffs during its timestop.
I'd just like to say, a Great Wyrm Red Dragon who has time stop available to them should be smart enough to know leave and not confront an enemy party on their terms.
If the party has shown up in the dragon's lair they are likely prepared for dragon slaying, people don't just randomly turn up the door step of the most ancient and powerful kind of dragon there is. I'm assuming the dragon has minions or magic that will allow it to know the party is coming.
The party should honestly end up confronting such a threat on what it believes are it's terms, prepared for such an encounter.
Ask the GM...
As the GM, do what you please.
Not even joking... it's your story to tell, so if you need/want the dragon's breath weapon to recharge, if it better fits the story you are trying to tell... absolutely it recharges during Time Stop.
Even if it wasn't supposed to originally, it does now, because it is simply better that way.
Maybe the dragon is beating the snot out of the party and you want them to catch their breath to make it a fair fight... so no, the dragon's breath weapon does NOT recharge during Time Stop. Screw you, lizard... calm down, the heroes aren't even putting up a fight...
Plan for both to be an option depending.