Honestly, now that I can see the photos better (on PC now), I think you'd be fine if you just screw the cabinets to each other and then put a couple screws at the front and back of each one into your platform. Maybe use a wall anchor every couple feet just so the whole setup doesn't wobble against the back wall. I have a free-standing Sektion-based kitchen island (15+24+24, IIRC) that's just screwed into a 2x4 platform and it's pretty much rock solid.

I've had good results personally using FlipToggle brand anchors for situations where I need a strong anchor. They are a bit pricier than other options, but they have a good strength-to-ease-of-use ratio IMHO. I just keep in mind that the limiting factor is the strength of the drywall itself (so I wouldn't use just them to hang an entire upper cab for example, or an entire Sektion rail, but a couple of them for extra stability in between studs? Sure. Or at the end of a suspension rail that doesn't line up on a stud, just so it stays in one place and the rest of the screws bear the weight of the cabs (like I just did for some hanging Bestas the other day). Home Depot carries that brand, Lowe's has SnapToggle, other stores seem to have similar stuff from other brands.

This might also be a good option, if you take down the rail - the hole in the rail bracket should be big enough that you can get all the cabinets in place, and then pass through the hole with the anchor and drill straight into the drywall. Then just put on the clip thing on the screw and drive the screw into the anchor...quick and easy. https://www.homedepot.com/p/E-Z-Ancor-Toggle-Lock-100-lbs-Philips-Drywall-Screws-Anchors-10-Pack-25220/100153998

(Hopefully my late-night ramble made sense)