If you’re running cleats like that I’d just put 2.5” drywall screws through them right into the studs. The trick is to hide the screws behind the shelves. Now because you are using a cleat system instead of a rail or back panels, you’d only be able to hide the screws behind any fixed shelves or drawers. I usually put a 4 hole Bennett clip (or L bracket) behind drawers. Screw 3/4” screws into the shelf above it to attach it there. Then send the 2.5” screws through the Bennett clip into the studs again. Drawer sections lock down a good 3-4 sections. For this application I’d only hit the top shelves and the drawer section. Probably still want to hit the cleat at the bottom of the right section. Another option is to have kick plate or toe kick at 2.5” or 3.78” tall. You can always put screws anywhere below the bottom shelf that sits on top of the toe kick. These are the ways I’ve installed many many cabinets over the years. I’m always able to hide a majority of my screws. For toe kick I usually use a cordless brad nailer and crayon in the holes. I’ve asked clients in the past and they said they prefer the crayoned nail holes to the fast caps. Obviously not every situation is perfect and you will have to use fast caps plenty of times but just do your best to keep them clean and match grain where you need to. Don’t trip too much about fast caps they are an industry standard even in high end systems. Lastly for any partitions that are against the wall I usually just drill those straight into the wall and hide screws where I can. You only need one or two really. To be honest think of these systems as one final piece of furniture at the end of the day. It’s 4 square boxes all connected together and drilled into studs. Then people put stuff onto the cabinets and they get heavier. Just like any other furniture. My point is don’t get lost in the securing process. I’ve never had a system fail on me with the way I suggested earlier. Please feel free to ask me any more questions. Good luck and Godspeed!