FCC approves giant space mirror as astronomers warn of risks to the night sky
(www.thebrighterside.news)
Also it’s who gets to approve such a monstrous plan?
The FCC. It's in the title. There's more detail in the article if you want.
Surely not a national government as it is an international issue.
Satellites have always been regulated by the nation they're launched from or owned by. There are international treaties regulating some aspects of that, but treaties are implemented by their signatories - the national governments.