Id just personally go with replaceable parts. AA NiMHs are like $1 to $2 these days, and 6V 5Ah lead acid is like $20 (Lol, first hit on google is $4 from some no-name brand).
Replace the battery every few years and you're set. Rely upon mass production, standard part numbers and highly recyclable parts (lead acid wins at this).
-------
Lead Acid is particularly good at UPS style power usage patterns. It's very easy to perpetually trickle charge lead acid.
------
If you're set on Li-ion, then use a standard 18650 cell, so you know that you'll always be able to buy a replacement. But given the attributes I see here, lead acid probably wins. So you have to replace a part every 3 years that costs $5 to $10, big whoop.