Regulated G2 is an under-explored backwater. Partridge of transformer fame proposed a series regulator valve just for G2 voltage. Not used much in the commercial world, but Grommes used it in their 260A power amp which has a 6L6 just for screen regulation.
Sometimes known as the 'Partridge control valve'.
With regard to power output in UL mode the valve isn't fully in pentode mode so power is a little lower, but you're trading this off for distortion.
As this is not a commercial product you can explore all avenues and have fun. You could even have a switch to select pentode or UL modes. Just don't switch it with the amp turned on