I was thinking of getting a power inverter LIKE THIS ONE and just using the charger that comes with the Air2.

For what you want to do - assuming you NEVER want to run more than that...

*(sorry i tend to edit a lot)

Battery

$139 https://www.amazon.com/Mighty-Max-Battery-Back-Up-Product/dp/B00OZMVI46

Take your pick of cheap PSW inverters

$45 https://www.amazon.com/BESTEK-300Watt-Power-Inverter-Adapter/dp/B07KQ4Q2L5/

You need input power of course

$103 https://www.amazon.com/Renogy-Monocrystalline-Solar-Compact-Design/dp/B07GF5JY35

and a charge controller

$50 https://www.amazon.com/Renogy-Adventurer-Charge-Controller-Display/dp/B015DDVDQS

~25-50 dollars for cables and other connection related gear. This will run a 100 watt TV for 3-5 hours each night (depends on daily input) without destroying the SLA.

If you got 2 panels instead of one panel you'd never have to worry about it being full until the battery finally dies. Probably need to replace it every 2 years.