18640 batteries (commonly used in powerful LED flashlights) are not hard to find online (although I've never seen them in a brick&mortar store), and hold many many more mAh. The flashlight I use on my bike requires one 18650, I keep two in rotation, as you describe. For my usage, I charge at most weekly in the winter. My light also can take three AAAs next to each other in a triangle, if it comes to that. The light had a plastic adapter shell to hold them, I've never tried to use it, and it is surely long lost now.

Some LED lights take CR150 batteries, which are also commonly used in cameras, and I have found them in drug stores before.