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We present a study of the dielectric, structural, and magnetic properties of the multiferroic or linear magnetoelectric substitution series [(NH$_4$)$_{1-x}$K$_x$]$_2$[FeCl$_5$(H$_2$O)]. Pyroelectric currents, magnetic susceptibilities, and thermodynamic properties were examined on large single crystals of the erythrosiderite compounds and detailed magnetic-field versus temperature phase diagrams are derived for three different substitution levels. With increasing potassium concentration the material is tuned from a multiferroic ($x \le 0.06$) to a linear magnetoelectric ($x \ge 0.15$) ground state. In contrast to the respective pure parent compounds with $x=0$ or 1, however, the ferroelectric or linear magnetoelectric polarization in none of the substituted samples is switchable by external electric fields, because these samples exhibit a significant electric polarization already above the magnetic ordering transition. The polarization arises at a higher-lying structural phase transition that is examined by THz spectroscopy and, on a deuterated pure single crystal, by comprehensive neutron-diffraction experiments. The structural phase transition is attributed to an ordering of NH$_4^+$ tetrahedra but does not break inversion symmetry in the pure material, while a finite K content causes pyroelectricity.