You may want to start by changing the port number on the FME license server and see if that helps:

https://knowledge.safe.com/articles/19460/specify-port-numbers-used-by-flexlm.html

Unfortunately the servers are mostly managed by a 3rd party and almost every change requires an individual service request. It took a month to get ArcMap installed and we've had this license issue now for 3 weeks.

Our department manages the ArcMap servers but not the FME server. ArcMap desktop is scripted to use port 27000 so all the clients will be affected if we change the production ArcMap license server.

One thought is to set up a test ArcMap license server on our test AGS server using a different port, submit 2 individual firewall requests, and then meet the people who have login rights to the FME server to test again.

This will take 2 weeks, and another 2 weeks to a month to implement production; hence the question. I don't have the option to put a non floating license on the FME server either. Maybe someone else has also experienced this before?

Another route I'm trying is that although the ports are open it may be that we specifically have to request connectivity among the 2 servers.

I can confirm that we run a separate Esri license server on 27000 and run a separate license server for FME Desktops on 27000.

You do say that it is your "FME Server", so presume you are talking about FME Server the software and that you must be using a pre-2016 version, as FlexLM isn't used for FME Server anymore

Have you confirmed that you can telnet to server2 on 27000 from the FME Server machine, just to make sure you can actually get through.

Is it possible that your network adapter is using IPv6? This is currently not supported by the floating license manager, so try disabling IPv6 on the client and see if that helps.