Gelatin-based hydrogels have emerged as a popular scaffold material for tissue engineering applications. The introduction of variable crosslinking methods has shown promise for fabricating stable cell-laden scaffolds. In this work, we examine promising composite biopolymer-based inks for extrusion-based 3D bioprinting, using a dual crosslinking approach. A combination of carefully selected printable hydrogel ink compositions and the use of photoinduced covalent and ionic crosslinking mechanisms allows for the fabrication of scaffolds of high accuracy and low cytotoxicity, resulting in unimpeded cell proliferation, extracellular matrix deposition, and mineralization. Three selected bioink compositions were characterized and the respective cell-laden scaffolds were bioprinted. Temporal stability, morphology, swelling, and mechanical properties of the scaffolds were thoroughly studied and the biocompatibility of the constructs was assessed using rat mesenchymal stem cells while focusing on osteogenesis. Experimental results showed that the composition of 1% alginate, 4% gelatin, and 5% (w/v) gelatine methacrylate, was found to be optimal among the examined, with shape fidelity of 88%, large cell spreading area and cell viability at around 100% after 14 days. The large pore diameters that exceed 100 µm, and highly interconnected scaffold morphology, make these hydrogels extremely potent in bone tissue engineering and bone organoid fabrication.