The minimum current setting could be due to the LM324 like nearly all OPs can not get it's output all the way to the output when there is a load at the output. Just a little of negative supply should do the trick. Alternatively one could add an offset to the divider, so that zero current could be shifted to some 10 or 50 mV as needed. No need to have the offset at the shunt.
Without a negative supply the constant current load would also not work well, when the output voltage is low. The simulation already includes a constant load before the shunt, so it would not interfere with the current measurement.
A LMP7705 or similar could be an option, though a little tricky to solder (VSSOP 8 case). As there is likely enough voltage to spare, I would consider a diode (like 1N5403) for the negative side current to drop something like 0.6 V from the negative side. Usually the laptop supplies are isolated, and thus no need to have the negative side of the input to be also the negative side (ground) of the output.