Sony A7 III (ILCE-7M3) is in the list of supportet Cameras, but not in the Menu for lens correction

It's a lens profile, not a camera profile. The camera body doesn't really matter (except as a broad category). They just make the lens profile with some base body model.

My a7r5 isn't listed anywhere, but most lenses come up automatically:

Some don't, and then I need to browse. Just set camera to "all":

Everything lists correctly once you find the lens:

Note that there is an embedded lens profile that will aready have been applied if you shoot raw. But with Sony, that profile just corrects CA, not geometry. For some other cameras, the embedded profile also includes geometry. In any case, it is highly recommended that you apply the lens profile in ACR/Lightroom, on the raw data - not in Photoshop on the rendered RGB file.