I’ve found this old issue: 3.4.4 Shape keys lead to incorrect shading GLES3 · Issue #61685 · godotengine/godot · GitHub
Apparently it’s the same problem. I’m using the compatibility renderer, which uses GLES3, in fact, when changing the test project to any of the other two rendereres, the issue is gone (even though the tangents are also modified by the blend shape, Godot renders the normals just fine).
In the issue they mention to export only deformation bones. I did so and the issue persisted. I just commented on the issue to see if someone can explain to me how it was solved, but I think it was not.