If you want a single FET CC source then you really need to change to a Depletion mode FET instead of an Enhancement mode.
Using a single FET however presents the problem that VGS(th) is likely in 1-3V range so the series resistor has to drop this voltage. This means you eventually simply run out of voltage to drive the LED.
In this particular application it may well be best to use a simple transistor based current mirror:
simulate this circuit – Schematic created using CircuitLab
To get a good current mirror you need a well matched (thermally and V(BE)) pair like the BCM847. The advantage here is that that Q1 will work until it reaches VCE(sat) of about 200mV, so the LED supply can drop to the LED Vf plus about 200mV and still regulate the current to the LED.
To read up on current mirrors you could start here.
It is also possible to use a pair of FETs to create a current mirror, and providing the VGS(th) is well below the 3.3V drive voltage this would work. There is good coverage of this here. I've seen FET current mirrors used to drive Blue LEDs with Vf only a few 10's mV below the supply voltage. Matched FETs tend to be about 10x the price of matched transistors which is a major disincentive to use them.
As an assumption:
- Your 6V supply is 4x Alkaline AA for example with new voltage of 6V and EOL voltage about 3.5V.
- The 3.3V supply is via a linear regulator such as the MC78LC33 with about 100mV dropout voltage.
- You are driving a Blue/Green LED such as this at 20mA with Vf about 3.2v.
The regulator can keep the 3.3V supply accurate down to a Vin of about 3.4V so easily copes with the EOL Vin of 3.5V.
The Blue/Green LED drive configuration cannot drop any more than 300mV from the EOL Vin so sensing current flow using a transistor Vbe is out of the question. The CC drive I suggested will work down to approximately Vf + 200mV or about 3.4V, so again easily within the EOL battery voltage of 3.5V.
The only viable alternative would be to use an Op-amp with a voltage divider from the 3.3V supply to sense current with less than 300mV across the sense resistor. The cost for this option is much more than the simple current mirror.