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.
it's true that a sufficiently fast moving object would be length contracted so much that it started interacting with the minimal LQG length, which would violate Lorentz invariance. Depending on how big the LQG loops are, that could be a fanstastically high speed that isn't achievable in the universe though.
I assume the paper she and Brian Keating are talking about were very explicit in how they tested this property and how Lorentz invariance was expected to be violated, so you could check the paper for specifics.