Can I use a 150W Iwasaki SDX white high pressure sodium bulb on standard high pressure sodium gear?
If so, should I use the S55 or S56 ballast?

Do you have a picture of the lamp? That might help.

SDX lamp won't work properly on HPS ballast. It needs a dedicated control gear to get a stable white light.

Thank you for the advice. Y’all are very helpful. 

The SDX will glow orange on a regular HPS ballast I think.

They definitely need dedicated gear. I'm in the not-so-enviable position of having a 150W lamp and no ballast for it plus a 100W ballast and no lamp 

Anyone want to swap..? 

Normally "white HPS" need a special ballast indeed (they need an active arc tube temperature control system), but Iwasaki designed and marketed white HPS lamps sufficing with "standard" European HPS ballast (for that power rating) and suffice with the passive arc tube temperature control it offers. But not sure if this this SDX is one of these or rather a standard white HPS needing that special stabilization gear.
But here I'm not sure, if the US ballast will be suitable, as US ballasts tend to have different load characteristics.
So even though the S56 matches the European HPS 150W at the 80V arc voltage nominal operating point, at different arc voltages they may differ and that with cause problems with the way more sensitive white HPS.

So best is to consult manufacturers documentation for the required equipment (whether it needs the special controller or not and on which supply,...).

Iwasaki made the Specialux for use with standard HPS ballasts. Philips also made the SON-T Deco, but it was short lived, since it takes 15 min to reach white light, and its color is unstable.

The Philips SON-T Deco was unstable precisely because they tried to make it work with regular control gear, and that was a spectacular failure.  It did not work, and was therefore quickly withdrawn from sale.

The key issue here is that all HPS lamps unfortunately have quite a wide tolerance on arc voltage, and hence power.  For the standard lamps, they can tolerate being slightly over-driven or under-driven without any great effects.  However the white lamps are driven much harder and critically close to the limits of their materials.  If the same tolerances were applied to their power, some would be so far over-driven that they destroy themselves in a very short period (as happened to SON-T Deco), or they would be so far under-run that they lose their spectral advantage.  Moreover, fluctuations in mains voltage can have a similar effect.

So if you are lucky and happen to have an SDX lamp whose voltage is close to the rated average, and you run it on a ballast and mains supply having excellent regulation, it will work perfectly with normal spectrum and full life.  But even small deviations from this will lead to short life or poor performance.  Because of this it used to be customary to use a power controller together with a magnetic ballast.  More recently since the invention of electronic ballasts having better power regulation, these do deliver much better performance with white HPS lamps.