Problem solved. I bypassed Family History Support and called the GSD directly. I opened a new GSD ticket, giving my name and stake unit number. Even though I told the tech that this was actually a Family History Center problem, I had to remind her a few times of that fact.
She told me that the ASA firewall I had installed in July was scripted incorrectly, so I had to go through the console cable exercise again. Maybe when we scripted the new firewall in July, I failed to tell the GSD that we had a FHC in this stake center, so they used the wrong script.
It took her several tries to Remote Control into the PC. The LANDesk Remote Control didn't work (wrong server). The ccn.notlong.com also failed until she told me to kill two processes through Task Manager.
She rescripted the firewall, then I rewired everything back the way it should be, and Remoteware connected to the IM server just fine.
Now I can rest ... until I drive to Nauvoo tomorrow for a funeral. Not mine, thankfully.