Within the mathematics of Special Relativity there exists the possibility of a preferred frame. Using the Lorentz Transform that exists in either LET or SR any inertial reference frame can be equally valid. That is the relative measures observed between inertial frames such as energy, momentum, frequency, velocity and wavelength will be indistinguishable in either LET or SR. This yields reciprocity between reference frames in either theory.
However, LET assumes that _internal_ reciprocity does not exist. That is there is a preferred frame and all others length contract, time dilate, and have non-simultaneity relative to it. These changes automatically add together in "moving" frames to make them completely unnoticeable by any known means. If ALL times or lengths in one direction are uniformly changed no differences would be noticed. The reasoning behind this is far from ad hoc but also not exactly testable by known means. For example you can use Einstein's light clock to show that:
t rate = (1-(v/c)^2)^.5 = time dilation factor. Either theory would derive this result. However, in LET only the moving frame's time is running slower.
Classically there are two Doppler shift equations (three if both the source and observer are moving relative to a medium):
fms = f / (1+ v/c) and fmo = f (1- v/c)
If time dilation only occurs in the moving frame then the frequency emitted from the moving frame would be slowed by t rate and the frequency received would be increased by t rate.
fms = f (1-(v/c)^2)^.5/(1+v/c) = fmo = f (1 - v/c)/(1-(v/c)^2)^.5 = ((1-v/c)/(1+v/c))^.5 = Relativistic Doppler Shift.
The math for all of SR, even Minkowski etc, can fit nicely into LET just like I showed above (even simultaneity). HOWEVER, there is no reason to assume a "preferred frame" unless evidence for it exists. Also trying to measure time dilation in your own frame when everything is slowed down proportionally is also not possible (likewise for length contraction).
Whether we live in a BU universe or a LET type universe is beyond current experimental evidence. It is metaphysics. (The only experiment to prove SR above LET would be communication faster than light that yields backwards time travel/ the only way to prove LET would be some fundamental breakdown in Lorentz Invariance at extreme speeds for mass or something similar possibly reveling a preferred frame). However, in either theory the results are identical for v < c so we DO live in a relativistic universe. Arguing which concept is "right" is not possible by any known means but since SR has fewer postulates it is the theory used.
There is still a lot of stuff we don't know about how the universe works.