Electrical – Charge car battery with lab bench power supply

There is an safety issue that should be mentioned. Car batteries can put out huge amounts of power. If you put together a project and you have a dead short somewhere you are going to start lighting things up. A fuse is a must.

That said, a battery can work if you can't get access to a regular DC power supply. It shouldn't be recommended though as no current limiting capabilities are provided. (this is why a fuse is a must)

Another note, batteries will vary in voltage. Don't be expecting a nice clean 12V supply

As I mentioned in my comment, your question is too broad for this format. However, in the interest of moving you forward in your project, I propose the question of: "What kind of power supply should I be thinking of building for this use case?"

A lab power supply is generally of the linear variety (as opposed to a switching power supply). I think the first decision here is whether a linear supply will be feasible for you.

That question comes down immediately to the question of heat dissipation. A linear power supply reduces its output voltage by dissipating the excess energy as heat. Worst case for the specs you mention would be 50A at short circuit, although 50A at 1V would be almost as severe. If your unregulated supply (transformer + rectifier + bulk capacitor) output say 15V, the worst case heat dissipation would be \$15V \times 50A = 750W\$. That's one heck of a big heatsink, even with a noisy set of cooling fans.

So I'd be inclined in your position to strongly consider a switching power supply which would vastly reduce the amount of heat dissipation required.

For that I think you'll need to elaborate your statement of your requirements and perhaps then ask a more specific question addressing the next step for you.