ESP's are rather power hungry, particularly when using WiFi. I can only recommend re-chargeable batteries (as really nearly all batteries should be) LiPo's are tricky to work with since they need special circuits for charguing and dis-charging protection. Voltage regulation is always going to be an issue which will go at the expense efficiency. I would go for 4xAA and regulate down, which can provide you with power for several days if you let the unit run continuously. Making it sleep will improve on this, but you still will lose some power at the regulator.