List of all FIR and their authorities
For posterity, I found what I needed here: https://www.icao.int/safety/istars/pages/api-data-service.aspx

They currently have 58 datasets. The one satisfying my needs is "Flight Information Regions - Names".

When registering for an API key, I am allowed to do 100 requests for free each month. Downloading the entire list counts as 1 request. So the way I read it, I am complying with their EULA. If I'm wrong, please comment.