LDO to regulate bootstrap voltage
In short, yes, that would work as then you would have a nice stable 12V. But your middle drawing (with the zener diode) could be edited for this to work. If you placed it in parallel with the capacitor, then you would get 10V there. From my experience it is a good idea to add a resistor in series too!

Obviously, if 10V is too low, then you can change the zener diode. (I am not 100% what voltages these come in but pretty sure there is one around 18V?)

Either one of these should do the trick. Easiest way is the zener, and if you decide to use a 7812 (could also use a 7815), then remember to have capacitors on the input and output, especially if you have a varying supply.