Seeing as how nobody else could copy/paste the youtube link, I thought I'd do it for you...
I'm still not 100% sure what the guy in the video did. It looks like he put the two field coils in series with each other, and then the armature in series with that... which is kind of silly, because they were already in a big series string, just in a different order. I think he might have intended to disable a field coil to reduce the resistance of the motor. Not sure. Of course, no matter what he did, there will be a significant loss of power, because the motor is simply not designed to operate on 12v.
The reason car vacs are 'doll house toys' are because they are designed to work within the maximum ampacity of a typical cigarette lighter outlet. Which... isn't very much. Typically 10 or 15A. Remember that the lower the voltage, the higher the amperage. Watts remain constant. Because of this, though, the lower the voltage, the thicker the wire needed to deliver the same wattage.
Incidentally, I believe low voltage motors also require thicker winding wires, making them heavier and more expensive than an equivalent mains voltage motor. (Also, most DC motors are permanent magnet motors, and magnets are heavy [though this makes them easily reversible]).
For example, a 5A vacuum motor at 120v = 600w. To deliver an equivalent power at 12v, you'd need 50A. Which is a lot of juice in car terms. Consider that an ordinary car's alternator's capacity is usually less than 100A. You'd also need 10AWG wire (~6mm).
However. Here's something interesting. A car's HVAC blower fan is not a far cry from an acceptable vacuum cleaner motor/fan arrangement. For inexpensive cars, they're fairly cheap too. You could easily buy one for 30 pound, and make your own vacuum... out of a bucket or something. You'd need a direct - but fused - connection to the battery, and have to be running the engine (and NOTHING else) while using it. But very much possible and I'm sure much better than the crappy car vacs available. Blower fans are usually only about 30A, however, so don't expect a Royal's capability.
As for an inverter... it *should* work. As long as the inverter is capable of outputting the vacuum's wattage. I don't think either 120 or 240v would be more advantageous than the other. Bear in mind inverters are usually sold by maximum output wattage, and not the wattage that can be reliably drawn constantly. They will usually put that number somewhere in smaller font size. I just saw a $70 inverter on amazon. Labeled 2000W. In the fine print, it says 'make sure not to use more than 1000w.' Again, the inverter would have to have a direct (fused) and solid connection to the battery.
I would also use a regular vacuum with a normal universal motor for that, not some complicated brushless DC motor Dyson or whatever (although they might technically have less issues on an inverter).
A universal motor will work just fine on any frequency, or even DC, and should not mind being on an inverter's imperfect AC power.
