Continuing my line of investigation, I've bought a couple of the cheap ($5 or so) mini-inverters that are abundant on ebay. Opened one up and it looks pretty wimpy, not much there other than one power transistor and a very small transformer:
Plugged it into my 12V system, the LED lights up. It runs a 115VAC 4W incandescent night-light bulb dimly. Tried a 15W CFL and it would not start. The mini-inverter emitted an audible whistle and its LED dimmed, but only an initial brief flash came out of the CFL.
And the point is?