Current draw different results
It sounds like your mA meter is dropping enough voltage that the load is not getting the 3.6V it needs from the battery.  Your power supply is being adjusted to overcome this drop?

One way around this is to use a current probe (clip-on type) with a scope.  They use the magnetic field from the wire to measure current through the wire, and avoid voltage drops experienced with meters.

It is an expensive solution, but it should solve your current measurement problems.