if resources are scarce its not for me to tell your company how to spend their money but a new server in prod to allow a side by side build would be really good, you'll want to refresh your hardware sooner or later. That will give you minimum downtime (and perhaps a cost saving?)

Putting the flexibility into the app code has got to be a worthwhile aim so perhaps use the test box as production for a while whilst you rebuild your production box, you'll have to isolate it as you cannot have two servers of the same name, and then migrate and repoint the app when its complete, hence the requirement for connection flexibility in the app.

As far as I can see the rename server would work if you go that way but as you can see its more than dropserver\addserver to be complete, its local groups and SSIS as well. I presume you are trying to minimise downtime so when do you rename the server, before or after the SQL install? Before would be better, no SQL issues then. As this is a total rebuild I presume you add SQL server level objects by script and atach the databases. That should be doable in a few hours.

Just some thoughts.