I may have missed it but why exactly is Lyris part of the ebonheart pact, even going so far as to referring to Jorunns as her king.
Isn't she a loyal member of the imperial legion? Why did she enlist in the ranks of what should usurpers and separatists to her? From all we know there were other options, for example the fighters guild or even Svagrimm available to her if all she wanted to do is help Skyrim.
So why would she ever join a faction that actively engages in the three banners war?
Well... maybe it's the same reason Sai Sahan recruited some random smugglers instead of taking the nearest mage guild to his school of elite warriors...