I think that with LET you prove POR, instead of assuming it as an axiom. In any inertial frame POR holds, because of the "conspiracy" of Lorentz contraction and "local" time dilation. Having done that you can then go ahead and assume it in your work. That is, you can use SR math and techniques. For example, transform to the most convenient frame for calculations (usually someone's rest frame). All comparisons will hold; for instance, the same twin is younger by the same amount. But you can then ask an additional set of questions, if you want (not necessary), about who's "really" moving, or contracting, etc. This approach makes it seem inevitable that in some extreme conditions, currently beyond experiment, POR will break down; but for practical purposes those can be ignored. There's an additional question, "how does the ether work?" It has to be an extremely unusual substance. I suppose one needs to simply assume "Ether works somehow" as an axiom, which of course Occam won't like.
It's similar to the situation in Bohmian mechanics. The math and physical representations are considerably more complicated than with normal QM approaches, but give equivalent answers in all currently testable circumstances. If for some reason you want to use it, the first thing is to prove that equivalence; then go ahead and use the normal, simpler, math techniques. But in extreme conditions, with currently unavailable test equipment, it makes different predictions; so if ever dealing with those, use the more complicated Bohm approach.
So I think (please correct me if wrong) with LET, after first proving POR for all accessible tests, presentation of classical mechanics as found in Landau - Mechanics would then go through without modification; with the understanding that in extreme circumstances it might break down.
