The mechanisms governing the dielectric blend properties at different length scales for phase separating blends with multiwall carbon nanotubes (MWNTs) are unravelled by tuning the microstructure. Thereto, compatibilization by interfacially segregated block copolymers (bcp) and random copolymers (rcp) of poly(styrene-random/block-methyl methacrylate) (PS-r/b-PMMA) was achieved in phase-separating blends of poly[(α-methylstyrene)-co-acrylonitrile] and poly(methyl methacrylate) (PαMSAN/PMMA) undergoing spinodal decomposition. In our recent work, we elucidated the effects of copolymer architecture and molecular weight on the percolating network of selectively localized MWNTs. Only short bcp and long rcp/bcp improved the connectivity and refinement of the PαMSAN phase laden with MWNTs and the resulting conductivity. In the present work, we study the effects of copolymer type, architecture, and concentration on the dielectric properties. We demonstrate a concurrent increase of the interfacial capacitance and decrease of the interfacial resistance of MWNTs with entrapped PαMSAN upon effective compatibilization. This is attributed to the increasing amount of connected parallel microcapacitor RC elements formed by the network of adjacent MWNTs enclosing a thin dielectric layer of PαMSAN. At high frequencies (above 1 MHz) the electrons hop between the neighboring MWNTs, whereas at intermediate frequencies, the electrons of the MWNTs tunnel through the barriers imposed by the entrapped PαMSAN. The physical characteristics of the microcapacitor network, namely the thickness of the microcapacitors and the volume fraction of entrapped PαMSAN contributing to the microcapacitor network, are estimated by describing the dielectric relaxation time and strength using the fluctuation induced tunneling model and the interlayer model, respectively. Combining the knowledge of the aforementioned parameters allows to describe the evolution of the total interfacial capacitance of the microcapacitor assembly as a function of copolymer type and concentration. Our robust and simple procedure to tune the MWNT microcapacitor network in polymer blends via the efficiency of the compatibilizer can be used to achieve a synergistic increase in the dielectric properties at different length scales.