Orthogonal Tchebichef moments of fractional order (FrTMs) serve as descriptors for signals and images. Many fields, including signal analysis and watermarking, have relied heavily on such moments. This study addresses three critical limitations in existing approaches: the computational burden of higher-order moment calculations, numerical instability affecting reconstruction accuracy, and orthogonality deterioration in large-scale signal processing. Furthermore, using the QR decomposition approach is crucial to maintain the orthogonality of the higher-order moments. We introduce an improved computational framework with three main scientific contributions as development of an optimized set of three interrelated second-order recurrence equations for normalized FrTMs, implementation of the Schwarz-Rutishauser algorithm as an alternative to classical QR decomposition methods, maintaining orthogonality with substantially lower computational overhead; and integration of these innovations into a comprehensive system for biomedical signal reconstruction and watermarking. The method in question was tested on two benchmark datasets the MIT-BIH arrhythmia and CHB-MIT Scalp EEG. The findings indicate that the proposed methodology exhibits significantly higher performance levels than current methodologies, with a 64.3% improvement in PSNR (reaching 147.08 dB compared to 89.74 dB in existing approaches), 89.7% reduction in MSE (0.0092 versus 0.09 average), and 84.1% decrease in bit error rate (0.25 versus 1.57) for watermarking applications. Processing time was also reduced by 64.3% compared to competing methods, making this approach substantially more efficient for implementation in Internet of Healthcare Things (IoHT) contexts.