This study examines the effects of kraft lignin, milled hemp stalks, and dicumyl peroxide (DCP) crosslinking on polybutylene succinate (PBS) composites, focusing on rheological, mechanical, and thermal properties as well as accelerated weathering and fungal performance. Two composite series were produced via twin-screw extrusion, (a) simple blends (B-series) and (b) DCP-crosslinked formulations (R-series), with emphasis on hybrid lignin–hemp composites (B-PLH and R-PLH). Rheological analysis showed that hemp fiber increased viscosity, while lignin reduced it, and DCP further enhanced shear-thinning behavior. Mechanical testing confirmed that R-PLH exhibited a 16% increase in flexural strength (42.6 MPa) and a 2.4-fold increase in flexural modulus (1785 MPa) over neat PBS, but tensile strength declined by 19%. Thermal analysis revealed a 14–26% reduction in mass loss rate and increased char formation (up to 16.3% in R-PLH), indicating improved thermal stability. Water absorption showed that hemp fibers increased hydrophilicity, further increased by DCP. Accelerated weathering led to significant color change and surface degradation, particularly in R-PLH. Despite lignocellulosic content, all composites exhibited ≤2% fungal degradation, indicating limited mass loss due to fungal exposure under conditions used in this study. Overall, B-PLH and R-PLH offer a balance of stiffness and thermal stability, though trade-offs in tensile strength and weathering resistance should be considered for sustainable applications.