You would probably be better served by putting the compost material in the bucket directly and skip the complex methods.
What happens if you leave the diverter in "compost" mode, and someone (sleepy you at 3am, a guest, etc) just starts running water? You'll end up with water in your bucket, potentially overflowing into your cabinet & ruining it and damaging the floor below.
What happens if someone has the diverter in "waste" mode and starts throwing large chunks down the drain thinking they'll go into the compost bucket? You'll end up with large quantities of mostly solid waste in your plumbing lines and end up with a clog.
Keep the bucket under the kitchen sink if you want, and to make it easier, devise a large handle that makes it easy to open the door with your knee or with wet, messy hands so you can open it and toss stuff directly into the compost bucket.
Leave the plumbing and the garbage disposal to water and the small quantities of solid waste they're designed to handle.