More thoughts:
If I'm seeing a voltage drop of about 4V across my 1M resistor in a 5V circuit, that means the AVR chip has a resistance of about 250K at that moment in time.
Using that resistance value I can calculate current consumption with a real working voltage, eg. 2.5V from a coin cell.
At 2.5V the chip would use about 10uA (assuming the resistance stays the same). This is far more than the datasheet says it should draw. The datasheet says it should draw less than 4uA at 2.5V in sleep mode with watchdog timer enabled.
(nb. This is a Tiny85V, not a Mega328P...)
I guess what I need to do is find a resistor which gives me a voltage drop closer to the 2.5V target voltage.