You can vent a toilet with 2" in your case. If the lav had been 2" downstream all the way for the vent that would work too. The sink or lav would be wet venting the toilet.
Have you pulled the bowl first and checked to make sure you haven't blocked the outlet of the bowl?
It sounds more like a blockage.
Sometimes the wax will drop off the bowl when setting and fall in the wrong place. Or someone used the toilet as a trash bin, and plugs it with broken tile or cardboard.
I would pull the toilet, bring it outside and try flushing it there. Then bring it back inside and try again.
Don't ever set a toilet in a bath tub unless you like to see chipped tubs and broken toilets.
It amazes me how many handymen will be too lazy to carry a toilet outside, they would rather risk chipping your new tub.