Any tips or clues on how?! I can't manage to defeat one at all.  

It may take a while, even up to a few days to beat one. You might even have to skip battling it entirely. The main factor is that your pet should be able to effectively harm the Etelia. I had this problem earlier on the event, and even if your pet can defend well, it won't be of any use. Try continuously using Blow, and only use Bite if necessary.

I use pets that have elemental attacks. Firebreath dragons, acid spit cockatrices, ice breath lykoses. They do maximum damage.