Measuring very low current
You can check your meters documentation, it most probably states the series resistance of the mA/A settings. It is most probably higher for the lower-current settings, and lowest for the 20A connection, which might be why this setting does not have a bad effect on your circuit.

That said, I doubt whether your problem is the voltage drop. Do you have a capacitor on the MCU side of the meter? An MCU will draw current ins pikes, which will not show on your meter (at best, they will be averaged). You need a capacitor at the MCU pins to supply this spiky currents.