Ok, after your 3" 90 degree elbow in the floor for your toilet sweep install a 3" x 2" wye and run the drain from the lavatory to it and then vent the lavatory and the lav sink will be the wet vent for the toilet. Now you still need to vent the lav. sink. in my area using IPC we can use an air admitance valve.
ps. you come up in the wall from the floor with the 2" drain for the lav. the drain needs to be 2" because it is also acting as the vent for the toilet.