A new family of single-ion-conductor block-copolymer electrolytes (BCEs), comprising poly(ethylene oxide) (PEO) as conducting block and poly(styrene sulfonyl(trifluoromethanesulfonyl) imide of lithium) (PSTFSI) as structural block, was developed recently. To evaluate the influence of the structural blockon the physico-chemical and electrochemical properties, we compare two single-ion-conductor BCE families with structural blocks made of either PSTFSI or poly(3-sulfonyl(trifluoromethanesulfonyl) imide propyl methacrylate of lithium) (PMATFSI). Small-angle X-ray scattering revealed that at temperatures lower than the PEO block melting temperature, the morphology of both families is lamellar whereas, at higher temperatures, the electrolytes are in a disordered state. Both electrolyte families present an ionic conductivity maximum for some weight fraction of the structural block (wBTFSI), named BTFSI. For wBTFSI > 0.17, the ionic conductivity of the PMATFSI-based electrolytes is larger than that of the PSTFSI-based electrolytes by at least a factor of two. Based on a detailed transport analysis, we show that the strong increase of the glass transition temperature is the main factor limiting the ionic conductivity. We also interpret the conductivity maximum of the PSTFSI-based electrolytes by a limitation in available free charges for wPSTFSI > 0.17 while the polymer dynamics slows down. The optimization of the ionic transport in this type of single-ion-conductor BCE requires promoting the compatibility of the Li+-bearing structural block with the conducting block.