Antique Cast Iron Sink Wall Mounting Support and Weight Issues?
I have removed several of these sinks in the past when remodeling turn of the century homes.

They were bolted directly to the studs with ¼” lag bolts that were ~2-1/2 – maybe 3” long. One had a bracket it slipped on to no screws through the back is this the type of bracket you have? that one had 4 nails into the studs.

There was no other support. 
All of the houses had lath and plaster walls at least ½” thick maybe slightly thicker.
One of the sinks still had the ceramic buttons that covered the bolts I believe these are the hardest parts to find originals for.

One caution most of the drains the drains were made from lead pipe and had lead seals. You might want to upgrade the drain to get rid of the lead.

Many of those homes had oak 2X4's but I don't think there will be any problem with pull out with fir or pine studs.