Just to prevent the next blown 440mA-fuse, as it is annoying to pay so much money for a new one:
Even with small current, you often also have to deal with "inrush current". The "MAX 0.6A" on your meter are misleading. You should be way under 400mA, to be on the safe side the next time. It might be tricky to estimate from A-measurement to mA-measurement, that it is safe, to change the test leads, so that you might want to use the record function in A-mode to capture some spikes about 0.4A, which does not survive your expensive 440mA fuse, before switching to the mA range itself...
However, I don't know if the record function of the Brymen is also able to measure 250µS to capture peak values like the Fluke 289 does for example?
That is the reason, why I prefer using my Gossen Metrawatt, when measuring current and yes, I don't care about the doubled used common, as it is precise enough for me and saves me a lot of money, as I will never ever have to buy a 440mA fuse again, because the Gossen Metrawatt does not have one! ;-)
Cheers!