Wire inverter to existing transfer switch or need another?

I've not dealt with that situation so I better defer to others on the xfer switch. I'd be looking to get an inverter with the switch in it, and wire that up after the existing switch and before the panel.

Those Li's should handle the hair dryer just fine without much voltage sag. Just be sure the 12v wires are big enough. I encourage folks to go to 24v for more extensive high-draw use, but that doesn't sound like your case.