I've used Loco2 before but I see that they're now called
https://raileurope.co.uk/
You shouldn't have any problems unless someone goes on strike. You are dealing with the French and Spanish here!
Carcassone airport is quite near to the station so a taxi would be cheapish, but there is a shuttle bus from airport to rail station for a couple of euros.
Unlike our rail system, French rail tickets are a similar in cost if bought on the day, so you could wait until you get to Carcassone. Ticket machines in the station are easy to use and have an 'English' button. You will need to change at Narbonne. It's a pleasant journey down through vineyards then along the coast across salt flats with flamingos!
There is a coach from Carcassone to Perpignan but I've never done that so can't comment.
Train run from Perpigan to Barcelona but you might have to change at Portbou. I've done that on a Sunday in the past as far as Girona without any problems.
A mate caught a Sunday morning coach from Perpignan to Barcelona a few years back without any problems but iirc, it was an early departure!
Enjoy!