Using a sample of >200 clusters, each with typically 100 - 200 spectroscopically confirmed cluster members, we search for a signal of alignment between the Position Angle (PA) of the Brightest Cluster Galaxy (BCG) and the distribution of cluster members on the sky about the cluster centre out to projected distances of 3 R₂₀₀. The deep spectroscopy, combined with corrections for spectroscopic incompleteness, makes our sample ideal to determine alignment signal strengths. We also use an SDSS based skeleton of the filamentary Large Scale Structure (LSS), and measure BCG alignment with the location of the LSS skeleton segments on the sky out to projected distances of 10 R₂₀₀. The alignment signal is measured using three separate statistical measures; Rao's spacing test (U), Kuiper's V parameter (V), and the Binomial probability test (P). The significance of the BCG alignment signal with both cluster members and LSS segments is extremely high (1 in a million chance or less to be drawn randomly from a uniform distribution). We investigate a wide set of parameters that may influence the strength of the alignment signal. Clusters with more elliptical-shaped BCGs show stronger alignment with both their cluster members and LSS segments. Also, selecting clusters with closely connected filaments, or using a luminosity-weighted LSS skeleton, increases the alignment signal significantly. Alignment strength decreases with increasing projected distance. Combined, these results provide strong evidence for the growth of clusters and their BCGs by preferential feeding along the direction of the filaments in which they are embedded.