Referring to the discussion in the Comments, assuming it is legal, and therefore safe, for you to continue.....
Backless with cutouts possible?
Well a coutout, yes, absolutely.
Because of the proposed installation location, one side of the cabinet (the one that will have the cutout) is hard up against the return wall. So your side can be screwed directly to that wall, using as many fasteners as you personally feel are needed1. Once this side becomes a fixed element, most of the normal reason for having a back immediately goes away, even with the amount of space the back takes up being very modest2 a back is just not needed here.
You do need to think about how you will hang the other side of cabinet since this isn't a conventional installation. While I suppose theoretically you could use keyhole hangers I'm not sure about how fiddly this would be in practice given the other cabinet side. So I propose you use L-shaped steel brackets which are utilitarian, but strong and cheap.
They do take up some space though, figure you'll lose max 3mm from the back and side where each is located. But since they don't need to be placed anywhere in particular, and in keeping with the theme of maximising internal storage, site them where they're most unlikely to interfere with items placed on any shelf — up high, just under the cabinet top and right below one shelf.
1 Given destructive testing data (from manufacturers and from comparative testing done on YouTube channels such as Gosforth Hanyman), I would be happy to use only two, but feel free to triple that number if you need to in order to feel confident. One rarely has cause to regret going stronger on a build!
2 Since you would be using 6mm MDF if you did fit a back you'd lose at most 10mm of internal space front to back, which doesn't seem excessive. Although perhaps you're very tight for space with the proposed contents, where every millimetre counts (I think we've all been there).