Ralph's last description sounds like he was trying ZZO's LDO
solution.
A 3 terminal current source is the easiest solution. namely the LM317ADJ
TO-220 devices can be used in a breadboard by soldering the pins to short resistor wires or a DIP header.
LDO's are thermally protected and useful for current sources and will not dissipate much heat if you limit the excess voltage drop from the battery.
Resolution:
Resolution on R increases with current and tracing shorts is a transient condition so it won't drain the battery long. So you could use 100 mA from a Lithium Ion cell or only use 1 mA and a x100 amplifier, then you have 2 Ohm full scale for 200 mV.
Heat:
100 mA drop from 3.6V cell = 360 mW. The current sense resistor will have 1.25V across Vo-ADJ as it always has in regulation for this style of regulator. thus 1/8W is fine for a 1/4W resistor. To achieve this use 12.5 Ohms = 1.25V/100 mA
This thread will be solved when Ralph defines all the assumptions we do not know. Design Specs: have Inclusions , exclusions, "must use" components (if nec.) and a real-purpose to satisfy this effort.
If something does not work, take measurements with a DMM of everything and report that.