We had our domain name changed and for our single-server (no workers) we didn't feel a thing. I'd recommend backing up the server just in case. What about simply uninstalling the server and then installing it on the new part of the network and then restoring it with your backup?
Appreciate the feedback Toby. I have passed on your experience and suggestion. If I hear anything more about what they do or how it works out, I will be sure to post here.