I don’t think a car battery charger will work, it all depends on the battery charger. Some of the very old ones might work.Can I use a car-type battery charger for a power source?....
I have had great results using this site. https://jacobs-online.biz/transformers.htm
They have everything you need, nichrome wire, transformers, etc. The size of transformer you need will depend on how long your cutting bow is, long bows require more amps. I power my short bow (about 10-12 inches) with a large wall wart. I don’t break out the large transformer unless I’m using the long bow to cut a wing.
I use 26 gauge wire for the short and long bows and 16 gauge wire for forming. Bend the wire into what ever shape you need and cut away. Works great for servo pockets and such.