I think I cracked the code on why my license was failing on that one machine and none of the others, and this is very likely one for the weird and wonderful issues thread. Hopefully Jim or someone at JRiver can confirm that my theory is correct or not.
Recently (as of version 1.4) Network Manager (a popular Linux networking tool) enabled a new feature. To protect user privacy on public wifi, Network Manager now randomizes the MAC address of the wifi interface when scanning for wifi access points by default. It's a default that can be disabled, and some distros have disabled it, but Arch did not. As I noted previously most of my Arch machines didn't have any license issues; on examination all but one of my Arch machines don't use Network Manager or were configured before Network Manager version 1.4 (so had the old defaults).
My machine where the license kept failing had the new defaults and was picking up a random MAC address every time it connected to a different wifi access point, which seems to be what triggers the license failure (or at least I managed to reproduce a license failure that way). If the JRiver licensing validation relies in part on MAC addresses to identify devices, I think we've found the culprit (at least for my case).
The Network Manager release in question came out in August of last year, but hasn't trickled in to all the Linux distros yet, but has been in Arch since September. Unless the distros alter the defaults, I would expect this issue to start trickling into other distros soon enough. Accordingly the top poster notes that he has the issue with Arch based distros, but not with Mint (which is running an older version of network manager).
If anyone else who is having this problem on Arch can test my solution by adding the following two lines to /etc/NetworkManager/NetworkManager.conf
[device]
wifi.scan-rand-mac-address=no
[connection]
wifi.cloned-mac-address=permanent
That will stop the randomization and make network manager use the actual physical MAC of your device (after a reboot). I expect to have one more license invalidation, but after that I'm hoping it will stay put. Testing is appreciated.