I live off-grid. I'm not the electrical expert of the household and Gwiz will definitely know far more about this than me, but as I understand it an inverter converts battery power to mains power. So that means you need one or more big batteries to use with your inverter, wiring, battery terminals and a socket to then recharge your little battery. And how would you keep your big batteries charged? You'd either have to install a wind turbine or solar panels and some kind of charge regulator or take the big batteries to your nearest or most convenient power supply. Probably home. Have I got all this right, Gwiz and others?
If I were you, and was interested in taking a laptop down to the plot or using fluorescent lights as well as a bog-standard drill, I'd buy a smallish inverter technology generator. They're not that expensive these days, especially if you compare with the cost of having to keep buying short-life-span, tiddly rechargeable batteries.
Bear in mind that at start-up, your tools and other devices need significantly more than their stated wattage to kick them into action. I'm told it's up to double. So you'll need to take that into account, as well as pairofacres comment about cheap inverters.
Plus, I wouldn't leave the generator at the site. Someone would notice sooner or later. The theft of the generator itself might be trivial in comparison with the cost of repairing damage done to get at it. They're not enormously heavy. I'm a 50-year-old, short, fat female and can move ours around without too much trouble!