I received a Black And While 1.54 inch display module from Waveshare, and compared its current usage to the 3 Color model. The 2 color module is far more battery friendly than the 3 color module, using only about 1/4 the energy to do a screen update. It also has a partial update mode, which uses about 1/13 the energy as an update on the 3 color module, but the screen gets corrupted with the partial update, and wasn't much use for this project.
This makes the 2 color display look ideal for coin cell powering, but... When I started testing it under the lower voltage of a discharging coin cell, I found it stops working at about 2.8 V. Since the datasheet says the display works to 2.4V, I sent an email to Waveshare support. They responded in a timely manner, but their response was:
The voltage is for raw panel. You need to consider about the PCB as well.
To down the operating, you made need to change hardware, we don't recommend that.
Waveshare Service Team
It looks like there is something about the PCB that doesn't like the low voltages. Since the PCB is just a bunch of capacitors, 3 Shottky diodes, which all tested with a .23 V forward drop, an Inductor, 3 resistors, and an N Channel MOSFET (Si1304bdl), I decided to exchange the MOSFET with a logic level IRLML6344. The IRLML6344 has a better Rdson and Vgsth, but it didn't solve the problem. This is a bit odd, since the 3 color display has essentially the same components on the PCB, and it works down to at least 2.6V. I'm thinking that the COG controller for the 2 color display, IL3820, does not really work down to the 2.4V claimed, or I have a bad display.