I found a picture of the original ballast and it appears to just be a series choke/inductor running off 220v AC. It also looks like the lamps have starting electrodes.
If you're lucky it might start off 120v, try using a 150w incandescent bulb in series as a ballast. If it doesn't start try letting it warm up on the WH3 then switch to the incandescent ballast.
https://www.ebay.com/itm/354839049965? I have one of these from a different seller and mine contains a CWA ballast, with the ignitor disconnected mine puts out 200v open circuit and 1.3A at ~20V.
It's not that simple. The current is the important thing with any discharge lamp, which for a SOX lamp is usually 0.6 or 0.9A and the OCV/striking voltage is approximately 500V.
This is a spectral lamp though and all bets are off with these, it depends on the manufacturer's specifications.
I know, a 35w hps ballast will put out something like ~1.2A into 20V (but it would probably get pretty warm) and a 50w hps ballast will put out something like ~1.3A into 40v (although it might not start 2 lamps in series even with the ignitor)
I think these lamps are designed for 220v open circuit.