Tam,
Again: experiments that demonstrate the existence of absolute motion/ether, if valid, distinguish LET and SR because LET relies on the ether as the basis for length contraction and time dilation. I don't know how to state this point more simply. LET is not simply the Lorentz transformations - it's a fully fleshed out theory that includes the ether.
From the Lorentz transformation, the speed of light does not depend on the (inertial) frame of reference that is chosen to measure it.
Therefore, if LET admits the LT, LET should predict that the speed of light is invariant.
Therefore, LET should not predict that detecting an absolute motion is possible.
Therefore, if the LET derivation of LT is based on the existence of an absolute frame of reference,
then the LET must also contain another hypothesis that makes this frame finally unobservable.
In any case, your point does not point us to a way to distinguish SR and LET experimentally.
I am still open to discuss such a possible experiment, but I cannot describe one myself.
Assuming that both LET and SR agree on the LT, it seems to me that distinguishing LET and SR experimentally is not possible.
Suppose, for example, that the MMX experiment would deliver a 1000000 times better precision than before.
Suppose also it would lead to a certainty about an absolute frame of reference.
I would then conclude that both SR and LET would be invalidated.
They would be invalidated because both theories would predict the speed of light is invariant,
and the experiment would then have demonstrated the contrary.
It does not matter that a "absolute frame" was mentioned in the derivation.
Only the predictions of the theory matters.
I don't see any prediction of the LET that depends on the preffered frame hypothesis.
Conversly, we can also conclude that SR does not rule out the aether hypothesis.
SR simply makes the aether hypothesis "superfluous", as it is well known.
It is superflous as far as the LT is involved.
In this sense, the SR derivation by Einstein is a cleaned version of the LET.
It goes without saying that SR brings more than that.