The East Empire Company is a funny bit of business and there are several factors to consider.
1) That little bit of nastiness with the Shatter-Shields. The question here is whether or not Ulfric would do anything about a Stormcloak supporting shipping company that has probably contributed to his campaign. On the EEC's side I would be wary of doing business in that regard, but in terms of Ulfric I can't say I would blame him if he looked the other way.
2) We don't really know how intertwined the Empire is still with the EEC. Yes, Vittoria mentions about trading with the Thalmor but well.. I think we can both agree her plans are irrelevant in the long run, spoilers not withstanding. It's also not outside of the realm of possibility that the EEC still has exclusive trading rights to certain items, like they did in Morrowind for ebony. If those exclusive rights prove to be more valuable than trade in Skyrim.. then, well, as a business they will go with the money.
3) The EEC still employs Legionnaires, or at least outfits their actual guards as such. The sight of people like Adelaisa Vendicci walking about has a good chance of inflaming sensibilities.
Furthermore, I don't really see the EEC being the great equalizer in terms of coin in Skyrim versus the Empire. Short of accepting Thalmor gold, there's not really anyway Skyrim can command a treasury the like of which the Empire can. Yeah, they've got more places to spread the coin around but we've seen glimpses of the kind of coin they have; at the conclusion of the Great War they were able to give large sums of Septims as compensation for the war.
I do concede though that the Empire's focus, and coin, is likely going to be the south of Cyrodiil not the North in the event of a Tullius' victory.