Consider running a 4 -6 gauge automotive grade copper fused wire directly from your starting battery inside to a pure sine wave inverter capable of handling your tool power charging station. Depending on the cordless tool brand you use, most have such an item.
The reason its better to run directly off your starting battery versus the built in 110 or 12v plug is the amperage limitation and thin wiring from the factory.
Google or YT the topic for better details. You could also go on the lawn care forums and do a search as there are “green” mowing companies that face the same issues.
A very old (2017) post in an equipment forum said this:
“We have inverters in a lot of our service vans that have tool batteries plugged in. No problems....but (there is always one) it is not the cheapy inverter. Our battery supplier has cautioned me to not use the cheap inverters. It will shorten the lifespan of the batteries and chargers. A lot of the cheap inverters don't make true sine waves and a lot of chargers can not deal with poor input waveforms.” End of quoted post.
There are also amateur radio enthusiasts who need more power than the integrated outlets can supply and they have setups sorta like I described.
From site ham.stackexchange.com comes this post regarding powering a radio in a vehicle:
“…The best way, and the ways the pros do it, is to run a separate pair of wires from the battery to the radio. These wires must be rated for the maximum current draw of the radio (and then some, since every ham initially plans to install one radio and then winds up with 5) and both positive and negative leads must be fused at the battery. If they aren't, and come into contact with anything, you may short your car battery causing a fire.
There's another bonus of connecting directly to the battery - some cars shut off the cigarette lighter when the car is off, but if you connect straight to the battery, you have control over how the poweris controlled. There are low voltage disconnects intended for this purpose which will detect when the car is turned off and shut off the radio a fixed amount of time later, but which can be overriden to allow you to operate the radio without turning on the car if needed.”
(End of quoted portion of post)
Good luck with it. Always fuse to protect the wiring.