I do not need any current limiting resistor, because at 2.4V the LED draws 10mA.
This might be a problem. If the LED only draws 10mA with NO limiting, it is barely functioning, at all. This is bad design, to begin with. That said, there is still something to learn and possible borderline success.
+1 amyk. She is correct.
I've tried it with a P-Channel mosfet, high side switching, and I get the same voltage drop.
You tried it with a 55A power FET with min Gate threshold higher than 2.4V, on average. You looking for a small signal FET.
With a Mosfet you need enough gate drive voltage to turn it fully on. That might be a problem at 2.4V.
You do not need to turn it on fully, in order to beat a BJT in this instance. And FET operate at 1.8V all the time. 50% of our modern computers have hundreds of thousands of FETs in the processor operating at 1.8V. You will easily beat the BJT. Any resistive load, no matter how high impedance, is going to beat the diode drop of a BJT in this instance. Since your LED voltge drop is approaching 2.4V, this means any resistive load will have voltage drop approaching zero, which is much, much less than minimum voltage drop of a diode junction in a BJT. (Also current will approach zero; but is enough to get your LED to light, I promise.... all these "approaching" things are why this is bad design. Running LED by voltage is like dividing by zero, lol).
Here is a small signal PFET, for example. For high side switching.
http://www.mouser.com/ProductDetail/Infineon-Technologies/BSS816NWH6327XTSA1/?qs=sGAEpiMZZMshyDBzk1%2fWixDt2IZOmxjILFS%2fF%252bTDmNA%3dYou can just as easily use NFET for low side switching.
At 2.4V, this has resistance of about 300 milliohm. This will not greatly reduce current thu your LED. Output impedance of your battery and wiring probably is greater than 300mohm.
So this is bad circuit, but given directly driving the LED at 2.4V works good enough for you, this will also work good enough for you. Drive it directly, not thru a BJT. To get fuller saturation. Guarantee this will work (as good as your original design intent, anyway. LED should never be driven by unlimited current source and controlled by voltage, only, unless by active control. Even the 300mOhm resistance of your FET is not enough resistance to make this a good circuit. Very small change in voltage will make significant change in LED output or even burn it out.) But bottom line, in this particular case, I guarantee an optimal (and cheap) FET will give your LED more current than any BJT in any configuration.