There's a lot to see in that corner. We spent 3 nights based in Esperaza and another 7 near Perpignan, in 2011.
If you head south from Carcassonne, through Limoux, you can follow the signs from Couiza to Rennes-le-Château. Park the car up near the tower and take a 2 min. walk to the church and site of the Abbé Saunière mystery (hidden treasure, Grail, etc.). Can be a short 30 min. visit. The little church is worth it, the museum not. There is a hole to see, in the floor of the little shop/ticket office - where someone broke in one night and used dynamite to search for the Abbé's gold.
Head back down into Couiza (5-10 min. max) and turn back towards Carcassone again for a minute, cross the bridge and turn right on the D613. Turn right again (15 min. or so, if I remember correctly) onto the D14. It's a pretty road that skirts the Pic de Bugarach, supposedly the inspiration for Spielberg's mountain in 'Close Encounters of the Third Kind', and the site of the failed end-of-the-world alien rescue of chosen humans (during the apocalypse of December 2012).
True. Weird stuff in those parts. Our B&B's British-born host described it this way: "shake any tree around here and a nut falls out".
Peyrepertuse can be done in 90 minutes or so, Quéribus maybe a little less. The climb is a bit steep, but for just a short bit. Cucugnan is between the two. Although in "Cathar country", these two are not so much Cathar castles, but rather fortresses that used to guard the old borders that shifted between the Spanish/Catalan and French kingdoms.
Of the two, Peyrepertuse is the most dramatic, IMO. It sits on the crest, longwise. You'll see the back end coming in and around from the D14. The cut-off that winds upwards to the parking lot is taken from the other side. Both châteaux are within view of each other.
From Quéribus, head back down the other side, from the D19 towards the D117.