Floorplan reasonability check: would need to put a kitchen drainpipe in a concrete subfloor slub

I'm planning a remodel in an appartment, and the way I see the space would work the best is when I bring a kitchen quite far away from the mainlines. And layout dictates that sink drain pipe should go in a concrete subfloor, which I estimate will be 8-10cm thick.

The pipe in the kitchen-side can go above the floor inside the cabinetry, but it must be underneath the floor in the hallway-side (that is drawn in yellowish and approximately 2.1 m long).

I'm also not perfectly sure how low it can be attached to the main drain (it's vertical and plastic), probably some millimeters (at least) in height would be lost there. So, if use 4cm diameter pipe and lay in with 4-6 cm incline it adds up to the subfloor thickness. But it adds up just in pure math. Is it reasonable enough for the real attempt?

The reason I ask is I must present a plan for authorities now, and then it would be impossible to place the kitchen in any other place and I would hate to lay the pipe without proper incline just to fit in, and then install a sink drain pump.

Also, should I use iron, or plastic tubes are fine to be laid within a concrete slab?

Thank you for your time and suggenstions!