Thanks for the responses all.
The basement is not finished. The pipe going into the basement pictured above is the one closest to the camera in this picture below. The previous, and new, sink, will be to the left of that joist, you can see the water pipes. The five studs come down near the drain's entrance to the basement, so about 3.5 feet from the gray support beam you see.
It is hard to see in the original picture, but the drain also goes up, inside the wall, to the attic. I guess whatever new solution we come up with will have to maintain that. The sink, being in a peninsula not abutting the wall, some plumbing in the wall will be necessary. Maybe it can be in the new 12" extension to the wall there (original picture, with no drywall)-- added because the new cabinets will extend further along that wall.