I don't think it will work. The front offers little structure to support a sink, so you need side support.
I had to repair a sink mount for a customer where the granite installers did little to support the sink. I had enough room to install 2x4s on each side. You might be able to get 1x4s in there. They will have to be securely fastened in place, so you may have to use 1x8s so you can get below the sink to screw them to the sides of the cabinet.
You will probably need something like an auto jack to support it while you are working on it.