I feel your pain! I've battled Bastilla's Bastion for a dark fairy and Alicane Swiftarrow for a sprite at least 30 times each trying to get one of the fairies for a pet and no such luck. Others that battled with me got multiple pets but I didn't get a single one! Don't know what to tell you except your not alone!
I got a Dark Sprite from Foulgaze, it doesn't give a card, first time I defeated him.
Then I fought Lord Nightshade, no pet. Then on that DS Quest to collect Obsidian Chests, I fought Nightshade for the second time, and I got a Dark Sprite, gives a card, No trade nor Auction.
I was farming for a statue from Alicane Swiftarrow with my friend, and I happened to get a Sprite pet drop.
I suppose I'll be considered lucky and you're all very tired to hear the words "Just keep trying" but I'll have to say I'm a pet collecter, and how I got half my pets was from the Bazaar.
Tip: If you don't see a pet you want, keep clicking on the Pet icon button until it refreshes and you catch a pet you want at the Bazaar.
If you have plenty of Gold: Then, once you catch the pet you want, buy it and don't bother with the name. You can always rename for 500 Gold at the Paint/Dye Shop (sorry I don't know the correct name) in Shopping District of Wizard City.
Hope I helped!
P.S. Don't spend too much of your time on farming for pets, and if they don't give a card, it really is not worth your hours and hours, days and days of farming when you could be leveling up.
By the way, has anyone gotten that Fairy Queen pet from Smogger, boss of Katzenstein's Lab? It looks beautiful and I really want it :)