Looks like it is a structural as well as electrical part - those relays are held by it? Looks like aluminum but may be steel - try a magnet. To repair it, I would think about bridging the gap with a strip of the same metal, held against the part with a few small self-tapping screws or small nuts and machine screws.
I think I see a cable also connected to that battery clamp - is that correct? And this is the negative post, right? I doubt that bracket is carrying the starting current. Have you removed and cleaned both battery clamps and posts?
Scratch it a bit with a knife edge or file if you want to know what kind of metal it is. I think below that silver coating, it’s probably copper or brass. Not sure brazing or soldering is going to hold… Going to be hard to solder or braze that crack back together stronger than it originally was when intact.
I may be crazy, but I think I’d drill a hole through the crack near the top. Push the pieces back together before drilling to make the crack as narrow as possible. Then put a short bolt in, a flat washer on each side, and a nut on the back and tighten it down as a clamp to hold the pieces together. Then you can do your soldering or brazing.
My first choice would be to find that part either new or at a salvage yard and just replace it, though.
Actually, it may be the easiest to do and the longer lasting repair: just make a hole, get the bolt there and it will happily act like a bridge for electricity.
If it works, it would make sense to spray some paint atop to prevent corrosion.
at 3+ years on that battery, I’d be looking real good at it if you are having starting issues. I agree with @texases that the broken (cut? designed that way?) metal piece is likely not carrying the starting current and shouldn’t have much of an effect on the starting of your vehicle.
If the crack is preventing robust cranking, your shop should be able to measure a significant voltage drop across the crack during cranking. 1/2 volt or more. That’s a very easy test. What do they measure?
Please let us know what you finally did to fix the problem.
Did you check out replacing the part? I would investigate that first before trying to ‘fix’ the existing part.
Did you check for parasitic drain?
Is your alternator charging the battery correctly?
Repeated discharging will shorten the battery life-most likely your battery should be replaced.
If one gets to that thing with a multimeter, the good idea would be to measure voltage drop during cranking not only across the gap, but between the battery post and the clamp… and simply between the battery terminals - that two tests should tell what part is at fault right on the spot.