That's an idea that hadn't occurred to me. I can actually reproduce it by only removing the power for the e-ink display and not to the MCU. I think it's just a quirk of the carrier boards or something like that, but I don't know enough about the design to really debug it.
In my project I powered off my arduino (and the screen attached to it) completely by using latch and control the latch using a PIC mc. Even in sleep mode, arduino consumes quite a bit of power while PIC consumes relatively nothing while sleeping (datasheet says "20 nA @ 1.8V, typical" but probably more since you will at least WDT)
Do check the quiescent current on your 1.8V regulator, though. That accounts for a lot of loss that goes away when you use something like a TPL5110 to disable it completely.
