Can I drive (relatively) high power LEDs from a constant voltage supply?
When an LED is powered from a battery it is best to use a 50¢ linear constant current regulator (CCR) rather than a resistor to limit the current.  For current over 350mA you will need to use parallel CCRs.  A good choice would be an On-Semi NSI50150ADT4G. 

The CCR will hold the current constant as the battery voltage drops from full charge.  

A $5.00 350 mA Mean Well LDD-350LS CC driver would work well as it can be powered with an voltage between 2 and 28V with up to 95% efficiency.  Better efficiency than a resistor in many cases. Still the input voltage must be higher than the out as this is a step down regulator.  You could power a red, orange, or yellow LED with 2V. Use an LDD-700LS for up to 700mA.

A 3.6V Li-on or LiPo is still a very good choice for a battery powered LED.