As a JDM doesn't actually use the serial port (it just uses some of the handshake lines), and a USB/Serial converter isn't likely to even produce the correct voltages, it's not likely to work. You've also got multiple drivers, the program trys to access the port directly, and the XP driver intercepts this and feeds it via a driver DLL. This driver DLL is then intercepted in turn by the USB/Serial driver, assuming it will even allow single access to handshake lines. JDM programmers have major reliability problems anyway, and this is going to make things FAR less likely to work.
An ICD2 or clone 'should' work OK via a converter, as it's a real serial port programmer.