Good news and bad news:

Good news is that the balancing resistors nicely eliminate the 60 Hz hum to an acceptable level. Bad news is that I get around 0.5Vpp of hum at 120 Hz.

I was wondering what it was, crosstalk, bad ground design, etc. 
Amplification works super silent with a direct 5V DC supply, so I know that the rest of the circuit is very clean. But even with the 'humdinger' resistors, this 120 Hz hum reappears despite filament AC voltage beeing a very clean sine wave. This is a known problem that vintage Hi-Fi guys, who wanna use period correct partts, struggle with until today.

The true reason is that those old tubes have a very thin, fragile filament without much thermal mass.
The filament is basically "blinking" at 120 Hz (when fed with 60 Hz AC), modulating the current of the tube.
I wasn't aware of that, because I'm only used to indirectly heated tubes that don't have this issue.

I found out that this was one of the reasons why mechanical TVs came so late, because indirectly heated tubes were not wide spread in the era of battery radios.
However, there was a solution. Copper oxide rectifiers existed since WW1, and they have been used in the 1920s in battery chargers and battery eliminators (radio power supplies).

Perfect. They are low voltage, high current devices, can be perfectly used to rectify 12V AC to over 5V DC. I ordered two on Ebay for $20 and $60 (the only ones available today that won't cost over $300). One has 5 plates, one has 16 or so, but I can prolly remove some to avoid too high voltage drop. Current rating should be OK to light two 5V/0.25A filaments.

Together with a cap/choke the supply should be ripple free, at least good enough for not seeing hum bars on the image.