Does the FCC license Russian and Chinese satellites -- or the radio frequency used to communicate with them?
In general, the country with jurisdiction over the ground station would be the licensing authority, and would handle that in accordance with ITU guidelines. I think ITU also handles satellite comms in general, so I don't see how FCC has a leg to stand on if they just went to another ITU signatory. It's roughly equivalent to going to Cabela's for your fishing license after the WalMart sporting goods counter refuses you over some store policy.
ISTR there are also some international guidelines that kick in when no
effective national jurisdiction applies directly. (For example, you set up your ultra-high-powered radio jammer on a Somali-flagged ship and park it just over the line in international waters near DC, or somehow successfully set up a micronation then start a 500,000W 24 hour polka station on 121.5MHz.)