Main-chain liquid crystalline elastomers (LCE) of ABA block copolymer chains are simulated using a polymer bead-spring model to understand how their sawtooth tensile behavior is affected by changes in the degree of immiscibility between the A and B block as parametrized by chi N, playing a role akin to reciprocal temperature. It is found that the toughness, namely, the energy absorbed under uniaxial extension, increases with chi N, but its behavior depends upon a threshold value of chi N that marks the emergence of crystalline domains. Below the threshold, the toughness increases slowly with chi N and has an irregular dependence on composition, while above it the toughness increases more rapidly with chi N and depends on the volume fraction of the minority block. These trends differ from those found in a previous analysis of LCEs [Nowak, C.; Escobedo, F. A. Tuning the Sawtooth Tensile Response and Toughness of Multiblock Copolymer Diamond Networks. Macromolecules 2016, 49 (17), 6711-6721] where the toughness was found to increase linearly with chi N and always be the largest for the equimolar block composition. The uniaxial deformation behavior of un-cross-linked block copolymer chains is also studied to show that block immiscibility is insufficient to engender a sawtooth tensile response and that it can only emerge with the further confluence of appropriate network topology and chain semiflexibility.