Poly(indacenodithiophene-benzothiadiazole) has received significant interest because of its exceptional hole mobility despite its near-amorphous thin-film morphology and brittleness at low M_n. In comparison, poly(indacenodithiophene-benzopyrollodione) (PIDTBPD) has a lower hole mobility but is exceptionally ductile at similar M_n. Herein, we synthesize random indacenodithiophene (IDT) copolymers with varying amounts of incorporated benzothiadiazole and benzopyrollodione (BPD), which introduces varied degrees of backbone twist to each respective polymer system. This allows us to elucidate how the BPD monomer introduction leads to conformational and morphological changes that influence the crack onset strain (CoS) and hole mobility of these near-amorphous IDT copolymers and the rates by which each material property responds to sequentially larger BPD incorporation. Results of density functional theory calculations suggest that BPD introduction does not lead to significant differences in backbone linearity between the studied polymers, and grazing incidence wide-angle X-ray scattering demonstrates that the degree of crystallinity within thin films is not significantly altered. It does, however, lead to a more varied circular distribution of the hexadecyl side chains around the polymer backbone. With increasing BPD incorporation, a crossover point between CoS and hole mobility emerges. At this crossover point, a random copolymer with 30% BPD introduction displays increased CoS and an average hole mobility value equal to that of the PIDTBPD system, suggesting that hole mobility is more sensitive to torsion along the polymer backbone, while the response of the CoS is relatively delayed. The data also suggest that the increase in CoS with increasing BPD content does not arise because of differences in rigidity but because the more circular distribution of the side chains makes polymer chains with sufficient BPD content better able to flow.