There’s actually no more difficulty in giving a neo-Lorentzian (as opposed to Minkowskian) interpretation to the time dilation of muon lifetime than to special relativity in general. Which is to say, there’s no difficulty in doing so as long as you see no problem with an empirically undetectable privileged reference frame. (And I would argue that the is, in fact, no problem with such a thing - it is both a philosophically natural postulate and even a scientifically natural one, in light of the empirically well-confirmed violations of the Bell inequalities).

To give a rough explanation of what I mean: as far as I understand it, the decay rate is something that we would theoretically determine by solving the Schrodinger equation for the quantum state. You’d start with a quantum state representing a muon, and over time it would evolve into a superposition of the muon state and various decay product states; the decay rate would be rate at which the amplitude of the muon part of the superposition diminishes. (There’s probably other ways of calculating the decay rate, but they have to give the same answer.)

The relevant fact here is that the answer you get from this calculation depends on the initial quantum state, and the quantum state for a muon at rest is different from the quantum state for a muon in motion. So, in fact, the physics predicts that the decay rate *does* depend on the motion of the particle. The fact that this corresponds to the answer we get from the Lorentz transformation is just a result of the Hamiltonian operator in the Schrodinger equation having a Lorentz-invariant form.