Calculating LED current using ohms law
I'm driving the display with a buck regulator set to 2.2V for testing


Don't do that. An LED is not like a resistor. Once over a certain threshold voltage the current increases rapidly, so small changes in supply voltage and/or circuit resistance will cause a large change in LED current. The graph below shows some examples of LED current vs voltage. Here you can see that increasing the red LED's voltage from 1.7V to 1.85V (a mere 9% increase) caused a ten-fold (1000%) increase in current draw!     



You should set the power supply to a higher voltage, and limit the current with a resistor in series. The resistor drops the voltage difference between the power supply and the LED, resulting in a current flow according to Ohm's Law, I = V / R (where V is Vsupply - Vled). 

For this calculation You can assume that a red LED drops a constant voltage of ~1.9V, orange/yellow ~2V and green ~2.1V (which isn't quite true since the voltage does increase at higher current, but close enough for most purposes). If you want greater accuracy then you will need to measure the voltage drop of your LEDs at different currents.

Your multimeter may suck, but you shouldn't blame it for showing different current readings on different ranges. Most meters read current by measuring the voltage across a low value shunt resistance. If the voltage required for a full scale reading is the same (eg. 100mV) then the shunt resistance value must must be higher on the lower ranges. Since your circuit is very sensitive to series resistance, even the small resistance of your meter shunt is enough to change the current.

When you inserted a 100&ohm; resistor and measured the voltage across it, you effectively added a large value shunt resistance. The current then dropped very low due to the small difference between the power supply voltage and voltage drop of the LED. The answer to this problem is to keep the large value resistor in the circuit, and raise the supply voltage until you get a reasonable current draw.