Looking at HKJ’s driver tests, buck drivers are 65-95% efficient, depending on the voltage overhead and quality of the driver. So if you get a good driver this would be an efficient option. Mtnelectronics has a selection of buck drivers.
Linear drivers are going to be around 75% efficient at currents around .4A with a fresh battery. The efficiency gets better as the battery voltage drops.
For currents around .4A, using lower voltage LiFePO cells would be more efficient since the voltage difference between the cell and LED is smaller.