Thinking about the ideas:
1. Donating to the Temple.
This case I'm almost sure that Aldradach became a more interesting place, because all Color camps need that egg. Heavy City Guard presence around the Temple (or, some competent forces paid for it, like LK ), less presence on other places. This is a possibility to bring more subtle military actions to the city, from the Camps. Sneak-ins, ninjas, magic attacks against the Temple, pissing off all dragons, etc. I don't think that donating the temple causes peace. But, this is the safest thing in my opinion for us, it can please the Red (or at least I can imagine that he agrees), and also the Gold because of the balance. Also, the egg is way too white, boring, could use some painting; we should keeping it unharmed, yes, but not untouched by skillful artists
2. Giving to the weakest Camp.
This is a lovely idea too, because one of the camps must declare that they are the weakest, and convince the others (or some judge, like an Avatar), this is an interesting licit / coming out. But I'm afraid that this will be interesting only for a closed circle of people, and we (and 99% of Df people) will have no info or other exciting outcome from it, causes no extra in-character gameplay. I'm not sure that it will please the Red, because supporting the weakest is probably not his domain (and not sure he will agree this deal), but sure will please the Gold because of balance.
3. Sell it / licit / auction etc.
More money = more inertia and more fun, yes But, I don't think Red would accept this as he is not a merchant type.
4. Creating a "king of the hill" battle for it (new idea)
Egg is on the center of the battlefield, all interested camps can assemble a hit squad with equal headcounts, and let them fight for it. This creates more game, but we as LK are out of this fun, unfortunately. But Red loves war, so I can imagine that he agrees this.
Yes, it would be exciting to have an intern LK champion selection tournament!
About carrying the egg: we can ask the Red that if he agrees, lend us an egg-bearer to carry it to the designated area.