So, money-spending ideas.
Furnishing a laboratory and library in Karak Eight Peaks could eat up quite a bit a cash, particularly if we pay for dwarf-quality equipment and don't hold back on optimizing it. It cost us quite a bit for just shoddy manling enchanting gear, getting dwarf specialists to build our new wizard's lab materials could probably eat a thousand gold or so. This is a direct investment in wizarding and would not even need a loophole.
Paying for classes in the Grey College at the estimated 100g/action slot will add up very quickly, and can easily absorb a few thousand if we want to master the rest of the spellbook, get the other five Enchanting perks, learn human-style runes/runecraft, learn power stone creation and use, learn ritual magic principles and whatever ritual spells the Grey College knows... we could easily be talking three to five turns of nothing but taking classes, or several times that if we're hanging out in the Empire and doing things other than self-improvement. I think it would be well worth the investment of money and time, because Mathilde should be putting time into polishing her skills as a wizard to the best they can be. Again, this is a direct investment in Mathilde's wizard skills which is permitted under the vow of poverty.
Paying off our debt to Stirland is obvious, and while it's not financially sensible from Mathilde's perspective, we do know that the administration of Stirland is really hurting for cash, or at least were recently. On that score paying our debt is less something to be done for Mathilde herself and more something done for the good of one of the Empire's provinces, so I think the Grey College would consider it a good investment.
Using the cash as a seed for expansion of military-related businesses (niter factories, etc.) would be a very worthwhile investment financially and in terms of producing useful industries in the Empire, though it would require a fair bit of Mathilde's time to actually start those up (and we might want to finish studying economics if we're going into serious trader's nonsense). The Greys would probably frown on this investment unless Mathilde can clearly demonstrate its utility for the good of the Empire and show how the profits from the business are not being used to give her a large personal cashflow, because Grey Wizards are explicitly NOT supposed to be building up personal income streams and "oh I have another legal entity for this" only goes so far. If we can sell it, though, there's a lot of potential here. We may wish to ensure that whatever we build/invest in reverts to the Grey College upon our death so that they have a vested interest in our success; Mathilde has no other heir, after all.
Using the cash as a seed for expansion of non-military-related business (casinos, etc.) would be a worthwhile investment financially... but ultimately I don't think that we can sell it successfully to the Grey College. It's just not accomplishing anything that the Greys want accomplished and unless we have "this is serving the Empire's interests" pretty clearly backing up whatever income loophole we want, it's not going to fly.
Investing in our fief is a poor choice unless we're just blatantly trying to burn cash. It's not a high expected return on investment and there's no reason that Mathilde would want to live there instead of the Grey College campus, her underground palace, or Karak Eight Peaks, all of which have much nicer things going for them.
Purchasing property with interesting magical properties (meaning high
ulgu concentrations or waystones or something) for eventual poking at or building a branch campus on could be worthwhile. The Haunted Hills have probably all been sold off by Roswita by now for a quick infusion of cash, but it's doubtful their new owners have had time to do anything useful with them or even do more than complete a quick survey, so if there's anything there we can still look to snap it up. The Grey College supports poking at magical things (as all Colleges do) so once again, as long as we have "in the event of my death property reverts to the Grey College" on the ownership papers somewhere, they'll certainly support this usage of cash.
We could go back to Barak Varr and roll for curiosities again. Spend a month wandering the markets and something neat would certainly turn up. Admittedly it might not be something neat that we could
use, but someone will be able to use most of what we find. And we might fish up useful enchanting components and such again.
Investing in a dwarf business might be possible with our level of rep. Imagine the possibility of, say, putting up the cash for construction of a heavy gyrocopter and then the pay for its pilot on a Karak Eight Peaks-(anywhere else) route? We know that they're already planning to do that using existing resources but being the owner of an aircraft could probably be a substantial source of income over time and a very convenient source of personal transit in a pinch. Ditto any other business, though those might be less fun. With Karak Eight Peaks being just now re-established, King Belegar would probably be very glad of any investment in his economy even if it comes from a manling. This is likely to be somewhat less objectionable than generic investment in the Empire's businesses to the Grey College because it doubles as a form of diplomatic outreach and lasting influence in Karak Eight Peaks, both of which are things that they want.
The Empire isn't actually a feudal society though. Importantly different is that the Empire has a standing army, not just a militia and some knights. We aren't responsible for raising troops, because Stirland has troops, that's what the army is for.
If we're not responsible for raising troops or providing military service personally then in what capacity does the Elector Count have the ability to order her vassals to do anything other than pay taxes (which are presumably what is done instead of providing direct military service, and used to support that standing army) and obey the laws (which will not include "Elector Count can order whatever they want" clauses because the nobility would rebel)?