Extrusion-based bioprinting imposes stringent mechanical constraints on bioink formulations, yet the rheological parameters governing cell survival during the printing process are rarely reported in a standardized way, limiting cross-study comparison. In this work, we systematically characterized the viscoelastic properties of alginate/methylcellulose bioinks incorporating strontium-enriched hydroxyapatite (Sr-OHAp) particles and Poloxamer 188, and assessed their effect on PANC-1 cell viability in bioprinted constructs. The power law consistency index K and pseudoplasticity index n were used as quantitative descriptors of bioink behavior. Addition of Poloxamer 188 reduced K by 52.1% in particle-free inks and by 64.6% in particle-containing inks, while n remained largely unchanged (≤2% variation), indicating that particles selectively modulate consistency without compromising shear-thinning behavior. On day 1, bioprinted constructs showed lower cell viability than cell-seeded scaffolds (53.9–58.9% vs. 96.7%); however, constructs containing Sr-OHAp (B3) displayed progressive recovery, reaching 84.0% viability by day 7, compared to 72.9% for particle-free bioinks (B1). These results demonstrate that Sr-OHAp particles act as rheological sensitizers that reduce extrusion-induced shear stress while simultaneously promoting long-term cell recovery, likely through their bioactive surface chemistry. We propose that systematic reporting of K and n indices should become standard practice in bioprinting studies to enable rational bioink design and consistent knowledge accumulation across the field.