In my experience, AA's never last long while Li-Ion's take longer time.
So even if you have spare AA's, you'll certainly need to use them.

To make a statement: I have a few camera's with Li-Ion batteries, running fine for almost 8-10 years.
My cellphone works already 7 years with a Li-Ion battery. Never had to replace it, but now I have to recharge the cellphone every 4 days instead of once a week.


A Li-Ion is rechargeable, don't see any problems. You don't even need a charger for the Edge Touring. Just plug it into the wall. I have a 220 Volt to USB-cable which does fine.
Even with a solar-usb panel you can load it while biking long routes.

I'd use it for just half day routes, where it won't be necessary to last for 17 hours.

Maybe you could opt for a Mio gps (Magellan in US?), they have both AA and Li-Ion series.

Anyway, if it doesn't rain this late afternoon, I go check out the gps on my bike for a 26km route after I finished work. I'll let you know if it works good or not. Hopefully yes.