The search for new materials for bone regenerative purposes is still ongoing. Therefore, we present a series of newly constructed composites based on β tricalcium phosphate (βTCP) and poly(3-hydroxybutyrate) bacteria-derived biopolymer (P(3HB)) in the form of 3D scaffolds with different pore sizes. To improve the polymer attachment to the βTCP surface, the etching of ceramic sinters, using citric acid, was applied. As expected, pre-treatment led to the increase in surface roughness and the creation of micropores facilitating polymer adhesion. In this way, the durability and compressive strength of the ceramic–polymer scaffolds were enhanced. It was confirmed that P(3HB) degrades to 3-hydroxybutyric acid, which broadens applications of developed materials in bone tissue engineering as this compound can potentially nourish surrounding tissues and reduce osteoporosis. Moreover, to the best of our knowledge, it is one of the first studies where the impact of βTCP/P(3HB) scaffolds on mesenchymal stem cells (MSCs), cultured in lowered (5%) oxygen concentration, was assessed. It was decided to use a 5% oxygen concentration in the culture to mimic the conditions that would be found in damaged bone in a living organism during regeneration. Scaffolds enabled cell migration and sufficient flow of the culture medium, ensuring high cell viability. Furthermore, in composites with etched βTCP, the MSCs adhesion was facilitated by hydrophilic ceramic protrusions which reduced hydrophobicity. The developed materials are potential candidates for bone tissue regeneration. Nevertheless, to confirm this hypothesis, in vivo studies should be performed.