It seems likely that an outpost was in Mournhold around the time of TES:III. Just because we didn't see it in-game doesn't mean it can't happen. Resources you say? Helseth was king.
Plus, this is for role-play purposes, the OP can have whatever they want.
It seems like what most people were saying before my post were arguments about why there probably wasn't an outpost in Mournhold. I didn't feel the need to repeat all of them, so I just summed up why it would be unlikely they established a presence after the events of Morrowind. As for it existing beforehand, I still find it very unlikely. You need to give more than just, "Helseth was King," for it to make sense. The Dunmer people were very xenophobic as a whole, and I doubt the majority of the people would smile on imperial presence in their capital city. In fact, I find Helseth as king to make it even more unlikely for them to exist there. He was the consummate politician, and had to balance the wants and interests of the Empire, with the wants and needs of Morrowind. Why would he waste political capital on allowing the EEC to set up shop in Mournhold? Regardless, even if he did, it would have been a difficult thing to achieve successfully.
And of course the OP can roleplay whatever he wants. That wasn't the question. The question was essentially, "would this be lore friendly?" And the answer is: not really. There's nothing that suggests there was a branch of the EEC in Mournhold, and based on the political climate of Morrowind, it's unlikely there was one we never heard about.