Any of the elements are equally effective; as far as I'm aware there are no elemental advantages or disadvantages on Beastkeeper. But having a powerful elemental attack can really help you win, as it will end the battle faster (and if the etelia can dish out 200hp per hit, you want to kill it quickly before it kills you). I like using lykos; they can have ice breath and still run another muscle gene like steel muscles or hound to boost their strength, which makes them great for both offence and defence. Ice spikes works great for them too, as it can give you a powerful rush that lets you avoid the etelia's ice breath.
Even my best pets sometimes get stalled by a tough etelia for hours, if not days (and I've only got a few pets that can reliably win against them, since I missed the kami events). I usually give it a couple of hours, and if they repeatedly lose, badly, then I call it quits and let them rest before exploring again. After all, I'll gather more crystalline ice by exploring multiple times and taking out easier etelias than by camping out by one at level 18 and letting it kill me for 3 days straight. I tend to avoid those that have silent pace if possible, especially if they've also got magic mirror.