If it happens on head phones it is NOT, i repeat NOT, a ground loop between your sound card and amp. The ground loop isolater is not the solution.
You are getting a ground loop between your PC/Sound card and inverter. Ignore all other components. Use headphones as much as possible.
This is what I would try: Ground the cassis of the inverter to the same point you ground the PC. Ground the Chassis of the sound card and chassis of PC to the same point. Ground the USB cable (if possible) to the same point. If you still get popping through the HEADPHONES, i would bet its an issue of inverter quality. Look into a DC to DC power supply to solve your headaches.
There may not be an easy solution to the problem. Again, these components are never designed for automotive use. Its not that the devices are bad, just not designed for what you are trying to do.