Yes, the battery solution works :)
As I don't have the copper oxide rectifier yet, I used modern diodes instead. Two diodes and a center tapped secondary winding of +/- 6.3V supply.
Without the battery, there is a nasty hum, which is even nastier after amplified by the HUGE factor of 8 :-D
Yeah, it was good back then. With the battery, the hum is gone - at least gone enough for the goal here.
The battery would look different in the 1920s, more like that: