Well, if you carve the tail, and then faint 3 times, then yes, you'll have the tail in your inventory. As I'm sure you're aware, you'll lose the money you spent on the quest, and any items you used. If, instead, you bring up the menu after you carve the tail and select abandon, you'll lose everything you acquired in the quest, including any carved parts. It's like you never started the quest in the first place.
So, to keep it simple: dying until you fail works. abandoning doesn't work.
Aside from that, may I suggest you start playing online if you haven't? I usually take lagi down in ten minutes or so with a good team. Good luck!