Biodegradable polymer systems based on poly(3-hydroxybutyrate) (PHB) and poly(butylene adipate-co-terephthalate) (PBAT) have attracted significant attention for fused deposition modeling (FDM)-based orthopedic applications due to their biodegradability, tunable mechanical behavior, and potential to reduce stress-shielding effects associated with metallic implants. However, the immiscibility of PHB/PBAT blends, limited melt stability, and poor balance between stiffness and ductility restrict their processability and functional performance. In this study, rheology was employed as the central design parameter to establish the relationship between reactive compatibilization, melt structure evolution, filament formation, printability, mechanical response, and degradation behavior in PHB/PBAT-based systems. PHB/PBAT blends containing varying Joncryl® ADR and MgO nanoparticle contents were prepared through reactive melt blending, followed by filament extrusion and FDM processing. FTIR analysis confirmed epoxy-mediated reactions between Joncryl and polyester chain ends, indicating chain extension, branching, and enhanced interfacial interactions. Rheological analysis demonstrated that reactive compatibilization significantly increased storage modulus, complex viscosity, melt elasticity, and relaxation times, particularly at low frequencies, indicating the formation of a more interconnected viscoelastic network favorable for stable filament extrusion and shape retention during FDM processing. Stress relaxation measurements further confirmed delayed stress dissipation and enhanced melt structural recovery in compatibilized systems. In contrast, MgO incorporation introduced rheological heterogeneity and altered relaxation dynamics through polymer-filler interactions and localized chain confinement. Mechanical characterization revealed a transition from brittle PHB behavior to ductile PBAT-rich systems. Among the investigated formulations, PHB/PBAT/J0.3 exhibited the most favorable balance between tensile strength, elongation, toughness, and filament stability, while excessive MgO loading reduced ductility and impact resistance despite modest stiffness enhancement. SEM observations demonstrated improved phase morphology and interfacial adhesion after reactive compatibilization, whereas MgO-containing systems exhibited increased structural heterogeneity. Thermal analysis showed that compatibilization modified crystallization behavior through chain branching and reduced crystallinity, while MgO influenced crystallization efficiency and degradation pathways. In vitro degradation in phosphate-buffered saline (PBS) solution at 37 °C demonstrated controlled degradation behavior and gradual pH evolution over 42 days. The results demonstrate that reactive compatibilization governs the viscoelastic state required for stable FDM processing and balanced mechanical performance, while MgO provides secondary control over stiffness and degradation behavior. The developed biodegradable PHB/PBAT-based systems show promising potential for additively manufactured orthopedic and biomedical applications where controlled degradation, flexibility, and processability are required.