If the wood won't work, HU-----35, ask your fabricator if he has remnants the same thickness as your stone, which should expand at the same rate. He might be able to cut lengths of stone to replace the wood filler pieces in my drawing. If the seam where the filler piece and stone slab is filled prior to installation of the backsplash, there will be less chance of water using the seam as a channel to the cabinets. Which is the only reason I can think of that HU----35's original idea wouldn't work--water getting through the caulk line and running down the backsplash piece. I would imagine that every cut and every seam will add to the final cost, so although I think this should work, will it be worth it just for the waterfall? Also, as I mentioned before, we see complaints about a lot of imprecise cuts, where counters don't match contours of walls, and these cuts would need to be exact to get the length you need for the waterfall edge.
You would need to have a serious talk with your fabricator, about whether your chosen stone can be cut at 45 degree angle without chipping and cracking.
Q
This might work short apron 6 9/16?
Q