Fibrous scaffolds capable of delivering natural drugs and herbs show great promise for tissue regeneration and wound care, particularly in personalized medicine. This study presents the fabrication and characterization of drug-eluting antibacterial core-shell mats composed of polycaprolactone (PCL) and pectin nanofibers produced through coaxial electrospinning. Berberine chloride (BBR), an herbal compound with antineoplastic, anti-inflammatory, antilipidemic, and antidiabetic properties, served as the model drug. Poly(vinyl alcohol) (PVA) was blended with pectin to enhance the mechanical properties of the core fibers. The shell was modified with two-dimensional Ti3C2Tx (MXene) nanosheets and subjected to covalent and ionic cross-linking. Structural analysis confirmed the successful production of bead-free fibers with diameters ranging from 160 to 350 nm, depending on composition. The PCL core fibers were uniformly coated with a pectin/PVA shell approximately 90 nm thick. The inclusion of BBR and MXene increased the fiber diameter. Drug-release kinetics, modeled by using Korsmeyer-Peppas, revealed a two-stage release mechanism. An initial burst release occurred within the first 24 h (kinetic exponent n = 1.36), followed by sustained release over 2 weeks (n = 0.48). The release mechanisms were identified as case-II relaxational release in the first stage, transitioning to quasi-Fickian diffusion in the second. Incorporating MXene into the shell further prolonged drug release. The mechanical strength of the scaffolds improved significantly by a factor of 7 and 4 in wet and dry conditions, respectively. In vitro biocompatibility assays using L929 cells demonstrated excellent cell attachment and compatibility. Additionally, antibacterial tests against Escherichia coli showed that the inclusion of MXene enhanced antibacterial activity by 30%. These results suggest that the functional biocomposite scaffolds hold the potential for developing innovative, drug-eluting wound dressings.