for me thats evidence enough, why would they have those things, living in a nord controlled area you think they would join the war with there hosts instead of keeping around the oppositions armor.
Maybe the owner of the club is a Legion veteran. That one man in Dawnstar wears his armor because he's proud of his service to the Legion, maybe the Dunmer feels the same.
A better question is this: Why would a SPY have Imperial Banners and Armor in his home? Spies are supposed to blend in, and having those things is about the furthest you can be from blending in in Windhelm.
And for the record, Nord does not equal Stormcloak.
I'd say it does, as it's difficult for Beth to make this reference with there being a proper quest based around it. Even if it wasn't true though, I really fail to see why Ulfric should effectively waste his resources aiding a people who seem ambivilent at best to his cause, with most Dunmer either being pro-Imperial or viewing that it isn't their fight. Arguably Ulfric could've gained their support by offering to take back Morrowind after his victory in Skyrim as it would've given him the support of the Dunmer refugees, but that may have angered quite a few of his supporters.
A note that you could pick up in the Cornerclub confirming that he was a spy is all you would need to include in the game to provide evidence.
And as for why Ulfric should help people who are ambivalent to their cause, why should the Dunmer seek to help someone who is ambivalent to them? The Stormcloaks are following Ulfric because he's promising a "free" Skyrim, and the rejection of the White-Gold Concordant, but those are things that are appealing to the nationalistic Nords. What is he offering the Dunmer of Windhelm, or any other minority for that matter? Nothing at all, so why should they WANT to help the Stormcloak cause?
As leader of the revolution, it is Ulfric's responsibility to rally people to his cause. If he extended offers to the Dunmer and other people of Skyrim, maybe they'd actually want to join in and help him.