How should this vintage wall-mount sink be mounted? Does not fit standard cast iron brackets
I am certain that if you mount that sink to the wall using only the two holes in the tabs, no matter how strong the mounting screws nor how hard the backing, the cast iron tabs will snap off. 

In the worst case, the tabs will hold the sink in place for days or even weeks, then break suddenly when you are barefoot in your underwear and the sink is full of soapy water and a wet cat. 

Don't even think about using this exquisite antique until you have figured out how to construct supports between the front mounts and the wall. 

Other than the thickness of the backing board, you have a lot of freedom to determine dimensions. The backing board should extend downward as far as conveniently possible to spread the levering force. Using the backing board also allows you to position the sink without regard to the wall studs. 

The screws and bolts that pass through the sink should be as large as will fit thru the holes, not only for strength but to avoid lateral motion. 

Here is a rough sketch of a possible installation: