I gather from your post that you understand it is never good to have a horizontal vent below the flood level of any fixture. You could simply install a new vertical drain in the wall behind the sink, elbow it, slightly off center, into the sink cabinet, and connect an air admittance valve before the trap adapter.
If you have 2 x 6 wall framing, install a 2" verticle drain to the left or right of the window, and connect a 2" teewye with the branch aimed towards the sink, about 16" above the floor. Reduce the top of the teewye to 1.5", and continue this vent vertically to at least 42"AFF. Extend the drain branch horizontally through the framing for up to 8' or as needed to reach the center of the sink............
The vent can run horizontally below the window as long as it finally intersects with the vent stack above the flood level of the fixture. another option would to address it as an island vent & trap below floor . air add. vents are bad choice. bob nyc lic. plbr