Get an adapter and try running your laptop from the 12 volt battery. Chances are it will run fine. If not, do as Mick suggested and get a Jaycar 12v PS.
Regarding the battery, I'd ditch the lead acid battery and get a 100ah lithium. Light weight and will run for at least twice as long as the lead acid, and without the linear voltage drop. Check out 4WD and caravan accessory suppliers for the most competitive prices on complete battery boxes with a lithium battery and all the ports you need.
You don't need an inverter.
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