Yup, all of the above.
When I was much younger and like all men, avoided reading the directions with an absolute diligence, I found an old pile of batteries in the bush, where an old logging camp used to be for about 20 years. All sorts of neat stuff was back there. I pulled out a lot of stuff for the scrap dealer. The batteries were a real gold mine for me. I had just gotten into shooting the new Lyman plains rifle in 50 cal that they had made up for them by Pedersoli. I still have it, 40+ years later.
To me, in those live forever, infallible days, lead was lead. They were still joining water pipes with lead solder and other things, like adding it to paint used on eating utensils and pots. What could go wrong??? A lifetime supply of lead was right in front of me. We had a hole in the back forty where we dumped old fence posts that were to full of staples and nails or just to rotten to make firewood. I threw all of those old batteries onto the pile of posts and set it on fire. I wanted to burn off the black rubber casings. Well that darned fire really got hot. Well seasoned 10 year old fir and pine posts throw a lot of heat when they burn, no matter how rotten they are.
The rubber burned off well but the lead all melted and sank through the ash to bottom of the hole. I shoveled the ash away and there it was. full of charcoal, dirt and who knows what else. Once the fire in the pit started, I left the area because it would take hours to burn down and several hours more before it would be cool enough to get near. Safety had little or nothing to do with it.
After I picked up as much of it as I could find, I had about three hundred and fifty pounds of lead. This was all thrown into an old cast iron bathtub for further processing. Again, a bunch of old fence posts, cut in half provided the heat source. The livestock wasn't overly happy as the tub was their watering trough. It was spring time and the creek through the pasture was flowing freely. Fussy buggers.
Again, I didn't stick around, again, not for safety reasons but because watching that much lead come to a melting point is about as exciting as watching paint dry. I came back often enough to add wood to the fire and scrape off the crud. Lots of fresh air and when you are close to that much molten lead, you stand up wind. I was lucky enough to have special hot purpose gloves, meant for handling hot glass molds and a good leather apron. OH, molten lead will find the little holes around the tongue of your shoes and seep through onto your feet. Socks do not provide any protection whatsoever.
I used a metal water dipper to ladle out the molten lead and pour it into old used bread pans. Heavy? Yes. To much? yes. I found that out the hard way to. Hey, give me a break, I was doing all of this on a try and learn basis. The only computers available at that time were in highly classified government and university bunkers. Books, were hard come by and certainly not available from the library. I used the knowledge I had learned from melting glass for bottles to melt the lead.
Anyway, the end product, which I later melted and poured into smaller, more easily managed blocks in cupcake pans and later into a cast iron stew pot, heated over a coal fueled brazier, oh how times have changed, and poured them into molds to cast .500 diameter round balls and .512 diameter maxi balls. Later, I used it for .359 diameterx162gr bullets for my SAA Uberti 357 magnum.
I will admit, the round balls were a bit hard to start, so I used thinner patches, made from old alter linen. The maxis were just about perfect and the Crisco lard used for lube seemed to do the trick. The projectiles were accurate and held together on target extremely well. I did recover a round ball on the far side of a deer, after it penetrated a shoulder blade and both lungs to become embedded in a rib on the far side. I never did recover a 500+ grain maxi. In the 357, I had to learn the art of sizing through a press and lubing with tallow/beeswax. Even then, I got some leading.
One thing I will say, that lead always stayed shiny. It didn't oxidize in the container over the winter like the pure lead castings did. It was much harder than pure lead as well.
OP, I tend to get a bit carried away with past experiences. This worked for me quite well. The recycled battery lead did the job nicely, didn't damage my firearm and because I had a fool's luck, I got away with my meanderings in melting and casting bullets.
The point of all this is, go for it. Use some common sense and take proper precautions. Recycle all of this stuff out of doors. Stay out of the fumes, wear a face shield. Wear proper apparel. Welding apron and gloves are just a start. Use leather covers over your shoes. If you are concerned about the fumes, wear a properly filtered breather.
Now, it never happened to me but a friend liked my lead and tried a similar experiment with the lead from two large 32 volt batteries he had. He used a large cast iron pot, over a propane flame. Very hot, very fast. All of the impurities reacted at the same time and started spitting little molten globules all over the area in a 3 meter diameter around the pot. Eventually it all settled down and all was well. His recycled lead performed well in his firearms as well. He never attempted to do it again though.
IMHO, stick to the right stuff the first time, every time. In the end, there is a lot less hassle or chance of injury and you have exactly what you want. Now, I only use wheel weights or pure scrap lead that I can salvage or buy at reasonable prices. My wife is married to a cheap bugger and I find a buck a pound is way to much to pay IMHO. If I am going to pay that much, I will just purchase already cast bullets from a commercial manufacturer, like the Bullet Barn.