BQ2057WSN Charging current issue
And after inserting the multi meter in series with Q1, the battery terminal voltage [increased from 7.39V to 7.44V] and the circuit was drawing 300mA from the power source [instead of 600mA]. Why so?


Because you inserted the multimeter between Q1 and the regulator's BAT sense pin.

For any typical hand-held multimeter to measure current, it must drop a little voltage, called the burden voltage.  This is what the meter is measuring and scaling to provide a current value output.  Dropping voltage slightly here fools the charger into thinking the battery is more charged than it really is.

Solution: place the meter at the 12v source, before the charger.