Best part is that you probably can't bring it back the way the bird flies without breaking it, you have to go on the twisty path back which is even longer :)
BTW how do you even get it back up to the camp? Is there a walkable path up (because I haven't found it I've always thrown my self down the cliff by the water fall).
Hey look at me, I had never played a MH game until this friday and now I'm talking like a pro ;)
OK, I just made an expedition to the shrine ruins, just to test the transportation of the wyvern egg. As you said, I cannot use the Palamute, which is a pity. But both wirebug and climbing walls covered with ivy works though. The climbing looks extremely awkward with the character holding the egg against it's body with one hand and climbing with the other, but I cannot deny the results. Both the wall at the wyvern nest and the wall before the camp are covered in ivy, so you can climb both. Faster way down or up is via wirebug, but if you calculate the height of your jumps wrong you break the egg. Climbing is slower and much more secure. So scaling the walls you actually can take the path the bird flies, which I did. I had to wait though, as Aknosom decided to land just at the next part of the path just as I was about to get there, so I had to wait until he decided to go, to avoid attacks which could break the egg. You deliver the egg at the blue supply box in the camp, just interact with it. I got 350 account points, which translate 1:1 to both money and Kamura points at the end of the quest. I also saw, that you can get a side quest to deliver transport items, which the wyvern egg is an example of.
So for another challenge without fighting you can try and deliver that. I recommend eating up enough rations to have full stamina, as you want to run as much as possible, as the egg slows you down a lot.