I took an hour this afternoon and put together somehow extended version of the converter, and moved from breadboard to protoboard.
In addition to previous simple DC/DC converter, this one has five components added
When turned on, voltage on LM385 is 1,24V, voltage on cap to be charged is zero, MCP6546 comparator "sees" more positive voltage on noninverting input, so its open-drain output is low. 7555 timer can oscillate, voltage on capacitor stars to rise; being more and more negative against circuit ground.
When voltage on LM385 versus voltage on capacitor is in ratio 1,5/2,7 - given by resistors on inverting comparator input - the voltages on both comparator inputs get equal and comparator output starts to rise up. Power supply of 7555 is cut off, transistor stay closed and circuit takes as little power as possible, mostly by current via LM385; being set to 25uA at 3V. As soon as capacitor discharges a bit, voltage on inverting comparator input rises, its output goes low, 7555 starts oscillating again and charging the capacitor again.
With components as per schematics, voltage on capacitor is stabilized at roughly 2,2V. Slightly lower than needed, but OK for experiments. Now I need to measure its efficiency and connect an actual supercap. Flux capacitor is to be added later.