[X] This is Dawongr Weber's money. The funds will largely go projects within the Karaz Ankor - your tower for one, but also investing in the Undumgi or Ulrikadrin, or purchasing goods made by Dwarven artisans or sold in the markets of Barak Varr.
[X] Karag Nar penthouse
Karak Eight Peaks is going to arguably be one of the most important trade and information locations in the world, so helping establish it further would mean a great deal for both the College and the Empire as a whole. It's an information nexus towards the East, a magical nexus that should be intriguing to the Colleges, a source of weapons from Karak Azul, in general a massive trade and economic boost for the world really, and notably is now also home to a great deal of human and halfing settlers.
The Karag Nar penthouse is an easy choice right now, as it's going to be the most in demand location that will be picked up if not taken. Even if we eventually decide to move our magical operations elsewhere, the great strategic advantages of this location mean that it would still be very valuable to have in our possession; and the sites that are likely to be more magically significant aren't likely to be in demand. The Grand Abyss likely won't ever be in demand at all, so there's little reason to take it right now.
[X] Convince Esbern and Seija to stay long enough to deal with the spiders in the Chiselwards.
[X] Write a series of papers on the magical phenomena you witnessed. Or better yet, get your Journeymanlings to do the actual writing.
[X] There's a growing concentration of royalty here. Get to know King Kazador, and this 'Sky-Thane' Prince Gotri.
I really can't argue more heavily for Esbern and Seija staying to help with the spiders. We know that these creatures are sapient, so the knowledge that they are currently starving and dying and likely to be exterminated very shortly is more than a little horrifying. These are potentially a species we can interact positively with, it's just we can't communicate effectively with them and right now are limited to just clashing. There are Amber wizards here though that can help greatly with this aspect, and even if the chance of peaceful relations aren't overwhelming it should still be worth it to try and make the world a little bit better rather than just dealing death, and notably, what are likely to be thousands of living creatures dying in the near future should take vast priority over two eggs. Thus this is my preferred vote for the Amber Wizards, with the benefit of them staying should give us capabilities that neither the King or ourselves possess.
The second option I've taken is writing the papers on the magical phenomena we've witnessed while it's still within our and all our journeyman thoughts and memories, along with the benefit of them all viewing the battlefield from different positions and having different senses so the overall report should be stronger than just writing down our own experiences. This is similar to the above likely to be a limited opportunity, or perhaps more accurately a diminishing opportunity, where we can provide great knowledge to the College of unique magical events and experiences so that it can be shared and taught more widely in the future. Notably this is also likely to get both Mathilde and our journeyman more College rep, which can be spent in the future for favors or magical items, in general improving the overall competence of our journeyman so they can contribute more, along with more college rep potentially meaning more Wizards are likely to visit. A plus to this option is that it can be enhanced by the Gold Wizard spells, so if any dice are going to be rolled, it's likely to get bonuses towards it.
The final option I was torn on, and went for the diplomacy option and interacting with the royalty. It helps provide connections for Mathilde with both people offering unique opportunities in the future whether that's reclaiming other holds or traveling via the air, and ideally I would hope that a meeting involving Thorek Ironbrow and Kragg would be involved as they are both important as foreign relations, and this is a limited opportunity where both are at the same place. I would love a scene where Mathilde and them both go see a waystone and see the ancient Dwarven runecraft on it.
I was very tempted to go with the Ranald Temple option, but with what happened there being unknown it's not really a limited opportunity given the location won't be claimed, and we can easily take it next turn.