Can the false front at the washbasin not be omitted entirely? The pipes should also be able to be placed inside the wall. 20cm is, in my opinion, quite a lot for a cistern; I'm pretty sure ours are narrower.
Can't you build the enclosure yourself? It maybe costs 40 € in materials and takes 1-2 hours and is really easy, even for those less skilled in craftsmanship. You get a nice hourly wage out of it.
That's just a 10 cm interior wall made of KS AFAIR - to chase horizontally for about a 50 mm drain pipe would be madness.
On the contrary, since it probably isn't load-bearing at that thickness. You don't chase, but rather cut out. Pipe directly into the wall. Insulation wool around it, if desired, mesh over it and plaster.