@stats20
I totally botched my above reply. I must have been distracted and saw your use of ":" and got derailed.

If you think you have a potential confounder you just add that term to your model, it is usually that easy. Like in your first model. However you seem to have a suspected multiplicative interaction and confounder. I have not dealt with this personally. I think you should draw out your model as a direct acyclic graphic and see if you have any content to direct your model. Does the confounder (W in lieu of COV) interact with only one of the terms or both? Perhaps Andrew Hayes or Tyler VanderWeele have written on this topic.

You could play the game of running a bunch of models to try and tease it out, but purest may call that data mining and undirected analyses. Of note, what if the confounder has a positive relationship with one term and negative with the other, will effects be masked or hidden?

A simulation may help with what to model or interpret results, but that would require knowing what the relationships are. If you find something, please update this post, I would be interested in the solution.