Why is my multimeter stopping device from booting when measuring current?
The problem will be in the shunt used to measure the current.  The resistance of the low current shunt is high enough that the voltage to the circuit drops when it tries to draw a (somewhat) higher current.
If you are only interested in the current after the circuit is started, then you could connect the meter as before - but short the meter leads.  This effectively takes the meter out of the circuit.
Once the start up phase is passed, remove the short.  The current will now go through the meter.
If the current draw is low enough, then the circuit should keep on going.