I did not expect the LCD to work in addition to the INA3221 power monitor on the I2C bus, but it does. I am powering the Arduino Pro Mini with an 8.4 Volt Lipo 2S battery on the raw pin, and an internal converter turns out 5 Volts, apparently with enough milli-Amps to run both my I2C devices. Very convenient during development. Here are some measurements of the current from the battery:

• with LCD: 39mA at 8.1 Volts, PWM=169/255, solar panel disconnected
• without LCD: 23 mA at 8.1 Volts, PWM=169/255 solar panel disconnected
• without LCD: 20 mA at 8.1 Volts, PWM=169/255 solar panel connected
• without LCD, with green LED on Arduino on: 29mA regardless of whether solar panel is connected

So the LCD draws about 15-20 mA, the green LED 5-10 mA. Without both, it is 20 mA.

The bottom line is, without bells and whistles, the board can do its job for about 160 milliWatts. Later: There are a lot more tricks to reduce power consumption of the Arduino Pro Mini.