Measuring the current of a small battery powered device
Do you really have to measure current?  A P-Mosfet like a DMG2305UX has digital switching levels and can switch something like 4 Amps of current.

Personally, I'd just slap down a P-Mosfet between the battery positive and the device positive (Careful! Drain to device as these things have parasitic diodes) and only worry about current if it gets hot or turns to smoke.