D'OH! My boss came back to work today, and pointed out something obvious I 
had overlooked. (not an unusual occurrence ..)

All these machines will be rebooted for Windows Update patching this 
weekend (and occasionally other needed updates, such as drivers, etc). So 
I can copy the script I have and the new file to the folder we keep on 
each machine for storing such update files. Then, when we go to run
Windows Update this Sun, we can just run the script locally, since we'll 
be logged onto each machine anyway. (we use WSUS, but only apply the 
updates manually, since most of our applications have their own specific 
shutdown order/procedure).

So I don't have to worry about executing the script remotely. Mind you, it 
means it pushes back my NW server rename project to next week, since the 
point is to have each client know the new name of the server before the 
server gets changed. But that's fine, too ... 

Now I just need to tweak the script to copy the new servers file a bit 
more robust and resilient, and write another script to copy that one (and 
the new servers file) to each client. But that shouldn't be too a problem.