I've not heard of anyone else changing server names and switching test and prod environments as part of an upgrade cycle. Keep prod as prod and test as test, then you won't have to change server names and devs won't have to change their code.

I can count the number of times I have changed a SQL server name on the thumbs of one hand.

There are a number of loose ends with changing a server name as you have already found, and I suspect to fully rename a server at the OS level you would have to go right down to the BIOS level.

Industry practice would be to put the flexibility in the connection strings, so keep it simple and push for that. You may be getting away with it now but start introducing complications like SSIS, SSRS, replication it will bite you sooner or later.