A final (at least for now) report on this:
I tried the another cable, and it likewise exhibited the problem on the one machine but not the others. Just for grins, I installed the appropriate Windows driver (which, also worked fine on other hardware) in a VM with the device connected to it (as a raw USB device, not as a COM port) and observed the same malfunction there, which assured me that it wasn't an issue that was common between the PL2303 driver in all versions of the Linux kernel and that particular chip on that particular motherboard.). The only thing I haven't done software-wise is try a bare-metal install of Windows on that machine to disprove any general problem specific to the kernel's generic USB driver (that only shows up with this kind of cable, and not any of the other half-dozen other kinds of Prolific cables & adapters I have, nor with the dozens of other kinds of USB devices I've used with it).
So, for now I'm just going to have to use another machine to talk to the radio. If I *really* want to program on this machine, I might buy an RS-232 cable, and try it with one of the USB/RS-232 converters I've used w/ this box, and see if I have any better luck with that. And if I happen to get my hands on a digital storage scope, I'd love to record the communication at the radio port and compare the signals.