Can I receive both an NSF international postdoctoral award and salary from a foreign university?
There will be rules that NSF sets for these fellowships. You need to look them up and/or talk to the program officer who administers the grant.

My best guess is that you can't double dip. So you will likely have to decline one salary or the other. On the other hand, there is nothing wrong with still listing the grant on your CV -- it is a grant you won in a competitive solicitation, so it belongs on your CV.