Is it possible to rewrite the Lorentz-transformations (for quantum particles) in terms of effective mass m* known from condensed matter physics?
Sorry to disappoint you, but the second "article" you are citing is the usual pseudo-science nonsense published by the crank crowd. I would suggest you pick up half a dozen good textbooks on relativity and study the introductory chapters until you understand the experimental difference between relativity and ether models. It's the only way to get past the mental block that exists in the human mind to getting over our evolved intuitive understanding for the Galilean metric, which, fortunately, is not how the world works (if it was, the world couldn't exist, to begin with!).
It takes time. In case of my personal experience it took several attempts at special and general relativity to "grock" the difference, which is profound.
Having said that, while many problems in condensed matter physics can be approached with techniques from field theory, the similarities of solid state systems to the physical vacuum are limited. The similarity of carrier transport in graphene to relativistic motion, in particular, is limited to a narrow energy range. In contrast, the consequences of relativity to the daily world (i.e. your existence in particular), require that relativity holds over an extremely large range of scales, which is probably well beyond 50 orders of magnitude.