I like the idea of the French cleat. However the challenge is now added thickness of the cleat itself. 3/4” may be too thick based on how deep the cabinets sit. 1/2” would be better. Thin, yes, but if I ripped an 8” strip, cut it at 45 degrees down the middle, secured it with 8 screws to the shelf, do you think the weight distribution would be sufficient? Again, I’m not trying to float the bookcase, just trying to alleviate the weight on the sideboard. Thoughts?