Maybe it's the hub? Have you checked their website for updates? Have you tried all of this without trying to shut down the inverter and amp with the computer? I'd imagine the amp should have its own way of shutting down, no?
When you're looking in the Device Manager, you're looking at the "USB Root Hub" properties, right? There should be more than one. They're most likely the only ones that would have the Power Management capabilities.
Maybe you could try unplugging all of the usb devices (gps, tv, hub) and just run the sound card straight in to the pc but using the inverter as external power, with no connection between the inverter and pc, no fancy wiring, basic setup, then test it out and how it goes?