Why would anyone direct connect a bar sink to the waste system? I think you should check your local codes and look up “indirect wastes”
By running an indirect waste to a vented and trapped fixture this would alleviate your venting problem IF you follow local codes and take the developed length and fixture units connected.
Most health departments frown about “bar sinks” connecting to a possible source of contamination of glasses and food utensils.
Oh well Another case for calling in a Licensed Master Plumber.
SylvanLMP