4 jaws are generally larger than 3 jaw chucks.
They don't suffer the same jaw stick out.
If you can find a manufacturers mark on that chuck you should contact the manufacturer. That doesn't look like a Chinese chuck.
That 4 jaw would be good to go, just get a backplate. I would recommend
https://cdcotools.com/ they now do sales on Amazon or ebay as well.. can't remember which.
I like their backplate, nice and thick.
For alignment on the chuck to the plate (3 jaw) see if you find registration marks (usually just punch marks that show alignment).