There's more points I'd like to raise in favor of Eike trying to be a Gold first.
Eike would have an easier time being a Gold Wizard and the EIC head, and be able to benefit more from that as a Gold than a Grey. With how the Golds have extensive customer bases and business partners, it's clear they are more openly prosocial than the Greys. Put selling luxury goods in contrast with us wearing the fact that we are empowered to kill people as openly as the grey on our robes. There's plenty of distrust for assassins, shady government officials, wizards, and spies, and Greys combine all that into one. That makes people nervous and hard to build connections with people, because they will reflexively have their guards up. But being at ease around someone who has the same goals and understanding as you is a lot easier. A fact a Gold can exploit much easier than an openly known Grey can. Winning friends, influencing people and getting them to spill intel is much easier when they don't think you're out to get them. And not being able to play politics is a weighty ball and chain as a business mogul. The Gold's spells are also much more useful to her than the Greys, even discounting the Grey College's restrictions. Law of Logic and Enchant Item would give her a leg up on the competition, as well as help her make the most optimal decisions for business and logistics.
Then there's the business side of things. The Greys strongly prefer being paid in favors over cash, and they are paranoid of what they do hand out. The Golds are all about alchemy, and are willing and able to sell the results to fund their studies. And where being a Grey Wizard means needing only fog and shadow to work your magic, Golds have much more intensive material requirements that being the head of a trade company alleviates entirely. Quantity and quality are going to be expensive, and her wealth can be applied in an approved and useful way in pursuit of learning and magic. Just like the Fey Enchantress, she would find using trade to get equipment for experiments on demand incredibly useful. Rare reagents from Lustria? Shipped in from Barak Varr. She needs multiple workshops built and tooled up? She could get it built by dwarves, and filled with tools made by dwarves at the expensive prices they charge. Chemicals, potions, and toxins in bulk? Name a price point, and she would be able to meet it as well as having it all shipped to her. Quality weaponry? Anything that money can buy. And we're well familiar with the value of having a lot of books. Easy access to resources will make her education and career easier and more successful.
Eike would also find more success, understanding, and acceptance rather than mere tolerance. Her dual loyalties would be able to symbiotically exist with the Golds, rather than being kept separate with the Greys. With the Greys, there would be distrust and suspicion because it's the Grey College. "What does this do for the Empire?" "Make sure you give up any money you can't spend in way that benefits the Empire." "Can you be trusted with this?" "Is your company a liability that you need to set down to better fulfill your duties?" Any success the company has needs to be gone over with the Bursar and approved by overseers, limiting her agency. With the Golds, there won't be any of that pressure. Success would be something to party over with friends, and she will have to answer to no authority save for paying tithes and taxes.
There's also her being able to build on the connections she makes in the Gold College by finding the right people for the right jobs inside the company. It already has many portfolios, so why not hire wizards to improve existing products or make new ones? She would also be able supply other wizards to improve the standing of herself and her College, as well as gain informal and formal favors with wizards and non-wizards. It would allow her to build bridges and friendships, a fact that Mathilde "Ruination of Camaraderie" Weber could stand to appreciate. She'd also get a lot of help in being a businesswomen, both with how the Gold Order works as a commercial enterprise as well as it's classes, and being in close contact with others who have similar interests.
And if you like Eike, why not wait until she's a journeywoman? If we're her master, we have to train her up and then send her off, only able to check in on her. But if we take her under our wing as a journeywoman, we can spend all the time we like with her and work side by side. It would lay the foundation for a relationship that lasts even after she becomes a Magister.
There's benefits to her being a Grey, such as us keeping an eye on her. But the College that gives her the most freedom, enables her to reach her potential, sets her up for success, and lets her lead her life to the greatest and fullest extent is the Golds.
