We present atomic-scale investigations of the thermodynamic properties of quinary bcc-based multi-principal-element alloys (MPEAs) within {Al,Co,Cr,Fe,Mn,Mo,Ni}, by means of short-range pair energetics and the cluster variation method (CVM) in the irregular tetrahedron approximation. We focus on an essential couple of key-properties detrimental for potential applications of MPEAs, namely the trends to (i) long-range ordering (LRO) and (ii) phase separation (PS). While the CVM has been more commonly employed in a grand canonical frame controlled by chemical potentials (δμ), we propose an original δμ-driven CVM scheme which allows an easier exploration of wide composition spaces typical of MPEAs. This modified scheme yields a slice-by-slice analysis of this space, by means of pseudo-ternary isothermal sections, which demonstrates the ability of the CVM to lead, with moderate computational effort, to full phase diagrams of MPEAs. In particular, our CVM calculations provide elements to interpret recent unexpected experimental trends of AlCrFeMnMo. Comparison with earlier studies of the same MPEAs using the less accurate point-mean-field (PMF) approximation indicates that CVM and PMF have global agreement for LRO at higher temperatures, whereas CVM should definitely be preferred as soon as PS begins to occur. Both approaches, employed more systematically in future MPEA studies in conjunction with ab initio electron theory and dedicated experiments, may offer convenient tools to check the merits of, and even improve, the various sets of effective pair interactions nowadays increasingly available for atomic-scale simulations of MPEAs, thus helping to identify composition domains relevant to avoid trends (i) and (ii) and design more reliable MPEAs.