How is Lorentz invariance handled in GR? [...] does GR imply small Lorentz violations in practice?
No. Lorentz invariance is independent of general covariance = diffeomorphism invariance. The latter only replaces the translation invariance of flat field theories.
Lorentz invariance is seen explicitly in general relativity coupled to a spinor field, where gravity is represented by a frame field and local Lorentz invariance is imposed on the frames.