Block copolyelectrolytes are solid-state single-ion conductors which phase separate into ubiquitous microdomains to enable both high ion transference number and structural integrity. Ion transport in these charged block copolymers highly depends on the nanoscale microdomain morphology; however, the influence of electrostatic interactions on morphology and ion diffusion pathways in block copolyelectrolytes remains an obscure feature. In this paper, we systematically predict the phase diagram and morphology of diblock copolyelectrolytes using a modified dissipative particle dynamics simulation framework, considering both explicit electrostatic interactions and ion diffusion dynamics. Various experimentally controllable conditions are considered here, including block volume fraction, Flory-Huggins parameter, block charge fraction or ion concentration, and dielectric constant. Boundaries for microphase transitions are identified based on the computed structure factors, mimicking small-angle X-ray scattering patterns. Furthermore, we develop a novel "diffusivity tensor" approach to predict the degree of anisotropy in ion diffusivity along the principal microdomain orientations, which leads to high-throughput mapping of phase-dependent ion transport properties. Inclusion of ions leads to a significant leftward and upward shift of the phase diagram due to ion-induced excluded volume, increased entropy of mixing, and reduced interfacial tension between dissimilar blocks. Interestingly, we discover that the inverse topology gyroid and cylindrical phases are ideal candidates for solid-state electrolytes in metal-ion batteries. These inverse phases exhibit an optimal combination of high ion conductivity, well-percolated diffusion pathways, and mechanical robustness. Finally, we find that higher dielectric constants can lead to higher ion diffusivity by reducing electrostatic cohesions between the charged block and counterions to facilitate ion diffusion across block microdomain interfaces. This work significantly expands the design space for emerging block copolyelectrolytes and motivates future efforts to explore inverse phases to avoid engineering hurdles of aligning microdomains or removing grain boundaries.