Did you download C2 from the scirra website and added it to your steam library or did you download it direcly with steam?
I don't know how you managed to get the overlay working with C2, I got a C2 pers. lic. with steam and I never got issues with the steam overlay.
If you want to dissable the overlay, simply:
1. Right click on C2 in your library in the "Software" tab
2. Click on "Propeties"
3. Uncheck the checkbox that says "Activate Steam-Overlay ingame"