I', trying to port a kitchen timer sketch from a pro mini to an attiny 167, just to see if it can go bellow the 4mA in sleep mode that I'm getting with the pro mini.
Did you see http://digistump.com/wiki/digispark/tutorials/proisp
As an alternative to trying different hardware, why not try to optimize the existing pro-mini. 4mA is far too much in sleep mode. You should get down to a micro amp or so. Some pro-minis have a trace you can cut to disconnect the power led and the regulator (assuming you are powering it at less than 5.5 volts). Also look here for ideas: http://www.gammon.com.au/power
And I just noticed that the code that I pasted is present in file C:\Users\%USERNAME%\AppData\Local\Arduino15\packages\ATTinyCore\hardware\avr\1.2.5\avrdude_conf.txt but I don't see nothing about that on the docs. Don't get me wrong, I think that Dr. Azzy is a real super-hero, but a small note about this on a easy to find place would be a nice thing.
You need the boost converter to power the display? It is surely responsible for a good part of the 4mA quiescent power drain,
I have several sensor circuits with ATmega328 running at 8MHz/3.3V consuming <50uA in sleep. These are mostly powered by 3xAAA NiMH or 1x 10440 or 18650 size li-ion batteries. I use mcp1700 or 1702 ldo regulators.
#ifndef TINY #include <LowPower.h>#else #include <avr/interrupt.h> #include <avr/sleep.h>#endif(...)#ifndef TINY attachInterrupt(0, wakeup, LOW); LowPower.powerDown(SLEEP_FOREVER, ADC_OFF, BOD_OFF); detachInterrupt(0);#else attachInterrupt(0, wakeup, LOW); set_sleep_mode(SLEEP_MODE_PWR_DOWN); cli(); sleep_enable(); sleep_bod_disable(); sei(); sleep_cpu(); sleep_disable(); detachInterrupt(0);#endif