Fragment shader normal artifacts appearing on specific GPU

I’m pretty sure I can narrow it down that: magFilter = LinearFilter on the height map interpolates among the RGB channels in a way that cannot – without creating numerical instabilites – be recovered from.
By setting it to NearestFilter, the artifacts disappear but with them also the smooth appearance of the normals.

Do you see a reasonable possibility to smoothen out the normals in this approach? Without having a smoothened bump map beforehand? I tried this before (using the approach described in https://github.com/mrdoob/three.js/issues/7386), without much success.