Well, she is undead; so, the Kemmlerites were possible a greater threat to her than many others would be.
This.
As to them being on opposite sides ... while Cowl was at the party, I don't think he was working toward war. I think he was working towards an infected member of the Winter court. So, them being in opposition isn't necessarily the case.
Regarding the Word, Harry said that it contained power over Mavra. Her wanting it off the market makes sense. Now if we see her with it again, we can think of other reasons, but she SHOULD have destroyed it immediately if she was keeping it out of others hands.