Cheating auto-shutoff of power bank
After some experimentation I have found out that increasing the on-time to slightly above 1 second (1.25 seconds) allows me to choose an off-time (11.75 seconds) much closer to the auto-shutoff time (~15 seconds). My guess is that the occasional failures with 1 second on-time are due to the on-time being somehow disadvantageously timed against the sampling time of the battery controller that measures current.

My best optimized version is now, on: 1.3 seconds, off: 11.7 seconds (that is a duty cycle of 10%, couldn't get it any lower), meaning total cycle time of 13 seconds. I have decided to keep some reasonable safety distance to the regular auto-shutoff time of 15 seconds. Otherwise slight variations in the parameters (due to e.g. temperature) might cause the circuit to fail occasionally/randomly.

With 10% duty cycle and 60 ohms I have now ~8 mA on average which means my 10 Ah battery will last max ~1.5 months which is quite okay.

See the corresponding Arduino sketch below. Although the values are likely to be very specific for my power bank model, I hope it saves somebody some work by giving a reasonable starting point. Remember that the Arduino software has to have the right config files for Attiny85 for correct timing and pin numbers.

void setup() 
{
  pinMode(0, OUTPUT);
}

void loop() 
{
  digitalWrite(0, HIGH);
  delay(1300);
  digitalWrite(0, LOW);
  delay(11700);
}