I have conquered the technology in my house. Bearing in mind that I no longer have in-house tech support (and, in fact, that the same tech support designed the system and that I didn’t really know what was involved), this is a Big Deal.
Email from said ex-tech support were involved in order to explain the system, but still I had to reset the puppycam, the wireless bridges, the main router, and the Audiotron. Then I reconfigured the Audiotron to use a local source for its internet radio stations (nice tool provided online here to create the necessary file — many thanks to Jon00 in the UK), ran all the scripts to change permissions, copied over all the files and restarted everything. Thanks to thebroomecloset for helping me test.
It all works! Everything works! And I took a fresh backup in case everything ceases working again! Yah me!