This little no-name 185Wh battery pack is $40 right now with the 50% coupon code and would probably work great. (I picked one up recently for another project and it actually does deliver as advertised performance.)
For another use case, I experienced that the phone battery starts swelling after few months. It was an indoor setup and I had removed the phone backcover to avoid overheating.
You can buy a 60,000 mAh battery (or build one) for about $50, which would buy this device 10,000 hours, or round it down and call it 1 year.