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Systems featuring hard-core-soft-shell repulsive pair potentials can form ordered phases, where particles organize themselves in aggregates with nontrivial geometries. The dimer crystal formed by one such potential, namely, the hard-core plus generalized exponential model of order 4, has been recently investigated, revealing a low-temperature structural phase transition, with the onset of nematic ordering of the dimers. In the present work, we aim to characterize this phase transition via a mean-field theory, by which a detailed analysis of the low-temperature properties of the system is carried out under quadrupole approximation. We determine the transition temperature and identify its order parameter, highlighting the link between the structural transition and the nematic ordering of the system. The first-order character of the transition is established and supported by the Landau expansion of the free energy in powers of the order parameter. The theory is subsequently generalized to take into account lattice vibrations and dimer length fluctuations. Finally, we provide an explanation for the anomalous behavior displayed by the specific heat in the vanishing-temperature limit, which is also supported by Monte Carlo simulations.