Composites of synthetic bone mineral substitutes (BMS) and biodegradable polyesters are of particular interest for bone surgery and orthopedics. Manufacturing of composite scaffolds commonly uses mixing of the BMS with polymer melts. Melt processing requires a high homogeneity of the mixing, and is complicated by BMS-promoted thermal degradation of polymers. In our work, poly(L-lactide) (PLLA) and poly(ε-caprolactone) (PCL) composites reinforced by commercial β-tricalcium phosphate (βTCP) or synthesized carbonated hydroxyapatite with hexagonal and plate-like crystallite shapes (hCAp and pCAp, respectively) were fabricated using injection molding. pCAp-based composites showed advanced mechanical and thermal characteristics, and the best set of mechanical characteristics was observed for the PLLA-based composite containing 25 wt% of pCAp. To achieve compatibility of polyesters and pCAp, reactive block copolymers of PLLA or PCL with poly(tert-butyl ethylene phosphate) (C1 and C2, respectively) were introduced to the composite. The formation of a polyester-b-poly(ethylene phosphoric acid) (PEPA) compatibilizer during composite preparation, followed by chemical binding of PEPA with pCAp, have been proved experimentally. The presence of 5 wt% of the compatibilizer provided deeper homogenization of the composite, resulting in a marked increase in strength and moduli as well as a more pronounced nucleation effect during isothermal crystallization. The use of C1 increased the thermal stability of the PLLA-based composite, containing 25 wt% of pCAp. In view of positive impacts of polyester-b-PEPA on composite homogeneity, mechanical characteristics, and thermal stability, polyester-b-PEPA will find application in the further development of composite materials for bone surgery and orthopedics.