Effects of renaming FC and using its original name for AG network name
An alternative you might not have considered is DNS. Obviously changing the connection string is easiest, but if that is a problem for some reason, you could leave the connection strings alone and remap using DNS. More importantly, think about these questions:


Why do you need to rename the server/cluster? 
Is the rename essential to business processes, or is it just a "tidying up" kind of thing?
Do you really need the geographical location in the name?
Will it really make sense to refer to it that way, since it seems logical
that part of the HA/DR topology might be to fail over to a different
geographical location?


In my experience most people have left that stuff alone, and documented very well that the original primary server was not renamed when they expanded geographically (or wished to rename for other reasons). You could also leave it the same and use DNS or the hosts file to let the applications and other servers/locations think the server was renamed, when really it wasn't. 

The actual rename is the hard part IMHO, and I would be looking at any alternative I could to avoid doing that. DNS, hosts files on all relevant machines, aliases, and changing connection strings are all ways you could bypass the actual physical rename.