Generator Transfer Switch for 2 Panels
No, those things cannot serve 2 panels.
Technically, those 10-circuit transfer switches are a very bad choice generally.  Aside from being insanely overpriced for what they do, they also have severe limitations, including what you're running into here.
This issue comes up a lot. I realize the devices have a consumer appeal to you, but I strongly urge you to re-examine your options.
Let's start with a photo of your two main panels, or at least, the model numbers of them, so we can figure out what other options you have.  Are the panels right next to each other?  Also photos of the listing of circuits on both panels would be good.
Is there a reason you DON'T want to make generator power available to most of your circuits?  It seems a great inconvenience only being able to power 10.