The melting-like transition in Li1Na54 and Cs1Na54 clusters is studied by using constant-energy molecular dynamics simulations. An orbital-free density functional theory technique is used, which scales linearly with system size, allowing efficient investigation of thermal behavior in these medium-size clusters. The range of temperatures covered by our simulations is from 50 to 450 K, and the total simulation time (for each cluster) is about 1.50 ns. This simulation time allows the extraction of statistically meaningful values for the thermal averages of several melting indicators (specific heat, diffusion constants, etc). The ground-state isomer of both clusters is shown to be icosahedral, with the Li atom located at the center of the icosahedron and the Cs atom occupying a vertex surface position. Orbital-free predictions for relative stabilities and structures of different isomers are contrasted with corresponding Kohn-Sham calculations in order to judge the reliability of our approach. The results show that both clusters melt in several steps, each one related to the activation of diffusive motion in selected groups of atoms. Nevertheless, only one broad peak is observed in the thermal evolution of the specific heat. Melting temperatures T-m, as measured from the specific heat peaks (at approximately 110 and 130 K for Cs1Na54 and Li1Na54, respectively), are different, suggesting the possibility of controlling the melting temperature of clusters by selective doping, and substantially lower than those of Na-55. Interesting differences in the melting behavior of both clusters, due to the size-mismatch effect, are highlighted. We also question the usual identification of melting temperature with the main peak in the thermal evolution of the specific heat. Several alternative definitions of melting temperature are then pursued. If we define, for example, T-m as the lowest temperature for which all atoms are able to diffuse across the whole cluster volume, very different values may be obtained.