How to close/insulate radiator niches in KSS-MW, Styrofoam, and gypsum?

Gypsum boards are not the worst when considered on their own. However, the description of the wall structure reads to me as a very heterogeneous Inslata Mista sandwich; I also prefer clean code in thermal contexts. How certain are your findings about the "old" wall structure?

The findings about the old wall structure are appropriate. I only detected the 2 cm air gap in some places through drilling, it might have been a coincidence as well.
Otherwise, the structure at the exterior walls is certainly like this:



It is a picture of mine, not from the internet ;-)

The findings on the old wall structure are appropriate. I only detected the 2 cm air gap in some places through drilling, it may also have been a coincidence.

That would, however, be a very unusual construction, both because of the finger gap of barely double the joint width and generally due to only stretcher shells.

Otherwise, the structure on the exterior walls is certain: [Bild] It is a picture of mine, not from the internet ;-)

Your thread 140262 in the Green section also strongly suggests that the wall construction differs between window sills and other exterior wall areas in your house (and is probably not limited to facing brickwork vs. plaster). The way shown in the picture here would be significantly more common.