Ordinarily, in switch-start fittings of that era, where the 5' 80w ballast also listed 8' 125w, this was in conjunction with a series capacitor. The 5' ballast in this setup, isn't showing this capacitor in the circuit, so must be getting that additional power from somewhere? On a normal stand-alone QS transformer, you'd expect to have just 4 terminals, T1 - line from ballast & connection to cathode 1, T2 - connection to the other side of cathode 1, T3 - connection to cathode 2, T4 - connection to cathode 2 & neutral. The QS transformer shown above, also shows a connection to L&N, so I guess it would be reasonable to assume it's more than just a bog-standard QS transformer. I must investigate the one in the fitting I got from you.

yeah its an interesting setup, I wonder if it contains a small autotransformer to boot the open circuit voltage to about 270V so then a shunt capacitor would not be needed?

Autotransformer would also be my guess, much better than a capacitor too which would usually have failed by now

Absolutely! Maybe I need to add mine to the restoration list after the Crompton Star quad and the Holophane Wallpackette I'm doing atm. Really need to get it out of the container and do something with it. Just a shame I'm really limited for places to install and use 8' fittings.