Vent/Developed Length upsizing?

[reluctantDIYer]

I want to add a kitchen sink/dishwasher to my basement. There is a 3" drain from the 2nd floor master bathroom right where I want to add it. But, there is no vent.

There is a bathroom in the basement about 50ft away with a 1-1/2" vent through the roof.

There is a nice big utility chase available to the bathroom vent. Can I run a 2" vent line horizontally & tie it to the 1-1/2" bathroom vent? If that won't work, how about a 3"?

Any help would be greatly appreciated!