Sabine is often right, but I think she's wrong here about Lorentz invariance being a problem, or at least a problem in the way she's saying.
Lorentz transformations are never going to length-contract the underlying fabric of space/spacetime. Relativistic length contractions contract moving objects, not the underlying spacetime.
In fact it's a strange and basic misunderstanding to have.
Depending on your reference coordinate system, space is transformed. That is the entire point of relativity theory. You might be misunderstanding things here.
Sabine is correct. All objects in a spacetime are anchored to that spacetime, so if spacetime has a minimum length, then length contraction of moving objects has a detectable lower limit, thus violating Lorentz invariance.