This is only a Watt (1) which any of these battery types will provide for a time.
If this is just a constant current circuit, then you need enough voltage to drive the LED + the overhead (dropout voltage). In this case only the rechargeable 10440 will do and the circuit will drop out of regulation fairly early in the battery discharge cycle. Most white (3W) LEDs will survive direct connection to a small rechargeable lithium cell and the CC circuit will have negligible effect on runtime, so I'd just leave it out.
A boost or boost/buck regulator would provide operation from any or all of the battery types.
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