What is the benefit of driving the HV5622 chip to the recommended operating voltage?
Arguing from the example of the CD4000 series logic line, I suggest that full voltage is required for full speed (8 MHz clock rate). Again arguing from other logic series (both the CD4000 and the 74HC) I'd suggest that running at 5 volts rather than 12 will reduce the maximum clock frequency from 8 MHz to something like 3 MHz, but that number MUST be determined by experiment. So you'll need to be careful of your timing.

I would also be concerned that the output transistors may not turn on completely (that is, that their Vgs(th) might be too high), but the article you linked to shows a working project. Nonetheless, you might keep an eye out for overheating and/or low Nixie brightness.