[SOLVED] LCD (1602, I2C) on ATtiny841

Finally, it works! Thanks for the pointers.

Final software setup with Arduino 1.8.5, installed libraries:
LiquidCrystal 1.0.7
LiquidCrystal_I2C 1.1.2

Test sketch:

#include <Wire.h> 
#include <LiquidCrystal_I2C.h>

LiquidCrystal_I2C lcd(0x27, 16, 2);  // set the LCD address to 0x27 for a 16 chars and 2 line display

void setup()
{
  lcd.init();                      // initialize the lcd 

  // Print a message to the LCD.
  lcd.backlight();
  lcd.setCursor(0,0);
  lcd.print(F("  Hello, world! "));
  lcd.setCursor(0,1);
  lcd.print(F(" LCD - I2C 1602 "));
}


void loop()
{
}

At just under 3 kB it would fit on a 441 even. Hope this is useful for others as well.