just find a 12v lead acid sealed battery like this for 20$, grab a cheapo charger also . at 4watts this battery would last a long long time before it drops below the 12v to run the device

https://www.amazon.com/Miady-Rechargeable-Sealed-Lead-Battery/dp/B0BRNB5QNR/ref=sr_1_4?crid=24PL98SIBWYEU&keywords=12v%2Bbattery&qid=1680750260&sprefix=12v%2B%2Caps%2C348&sr=8-4&th=1