Collioure, Toulouse, Carcassonne, or Perpignan before heading to Barcelona

I'd probably pick Perpignan. There is a direct train to Barcelona. 1 hr and 20 minutes. I am assuming you are renting a car for your castle tour.

Albi's about 45 miles northeast of Toulouse. You should definitely see it before moving farther south. I don't think it make sense to drive south toward Carcassonne to a 3-night base only to turn around and drive back to Albi at some point. Perhaps you can see Albi on December 30. Do check the hours of the Toulouse-Lautrec Museum in Albi to be sure it will be open.

I thought Toulouse was a very handsome city. It's a major university/business city (aerospace industry?) rather than a tourist destination. Keep it in mind in case you run into significant weather issues that make driving around the castle area not so enticing. I have a feeling you won't exactly be using freeways to get to those castles.

ViaMichelin gives nearly identical driving-time estimates for Toulouse-Carcassonne and Perpignan-Carcassonne. However, it appears the Cathar castles (which I haven't visited) are south of (and rather close to) Carcassonne. Perpignan does look like it's well-positioned. Or you could stay in Carcassonne itself. Or in Narbonne; it's closer than Perpignan to Carcassonne and some people prefer Narbonne to Perpignan. Narbonne is more prettified, I would say, but there is an interesting palace in Perpignan.

Carcassonne itself is central to both Albi (drive North across the Montagne Noire, it is quite scenic) and the Cathar castles. It would be my pick as I wouldn't want to change accomodation over three nights.

Ok it looks like the Gare de Toulouse has lockers and the Toulouse Latrec Museum is open on Mondays (closed on Tuesdays, May 1, November 1, Xmas Day, New Year's Day). I plan to lock my bag up and go straight to Albi from Paris and then maybe stay near Carcassonne

We visited Perpignan, it was OK, but my Daughter did Carassone and loved it.

I enjoyed Carcassonne more than Albi. I liked the BIG church in Albi but am not a huge Lautrec fan so the museum next door didn't interest me. After that, I didn't see a need to visit.

I camped walking distance from the old cite and only walked in at night because I was making day trips to see castles south of there. I'd recommend Foix as the best visit in the area. Others were a nice walk up the mountain and evocative - but in very ruined condition and probably very slippery when wet. The locals kept warning me to be careful of rain or lightning and come straight down if I encountered either.

North of Carcassonne the town and castle of Najac was another good visit. Cordes sur Ciel is an almost too perfect medieval town. It gets crowded in the summer (was fine when I visited in June) but shouldn't be at all crowded in December.

Perpignan isn't on my list. It seems it's only draw is as a transportation hub. I'd like to visit both Collioure and Narbonne but they weren't on my itinerary and December probably isn't the month to enjoy the coast.

Carcassone is the most convenient for the Cathars. Perpignan is the most direct for getting to Barcelona.
Will you have a car? Otherwise visiting the castles will be difficult as they are quite remote and aren't well-served by public transportation.
It is going to be very cold up in the hills where the castles are located...make sure to bring layers.