... well, the worst scenario is that you have too many flights landing at the same time, in that case you might have to queue a bit... but the waiting line is rarely maddening. There are an average of 1200-1500 taxis at the airport taxi ranks.
As per your trip to the airport at 3:30am there is no problem at all... just call one of the companies I mentioned. You can pre-book it say the day before or if you prefer simply call half an hour before you need to be picked up. 3:30am is not such a bad time for finding a taxi, you could even simply pop down to the street and if it's a major street you wouldn't have too much problem in finding a taxi passing sooner rather than later.
All of the above has of course a few exceptions... (1) depending the area of Barcelona you're staying in the availability of taxis circulating in the street at 3:30am can be limited (best pre-book then!), (2) on Fri and Sat, from 3am to 4am it can be a busy time since many party-goers start returning back home.
Your note on not accepting short rides... we share the problem here :)... luckily, the airport is 12km away from Barcelona so that's not a "short" ride (<15'), hahaha!
PS: Hotels Catalonia is a brand name of a chain so you'd need to be more precise for me to understand where your apt is located. Normally these hotels have an added 'name' to the brand (ie. Hotel Catalonia Barcelona Plaza, etc)