In this paper we study how distant weak interactions impact the dynamical evolution of merging binary black holes (BBHs) in dense stellar clusters. Recent studies have shown that BBHs undergoing strong interactions in dense clusters are likely to significantly contribute to the rate of observable gravitational wave (GWs) mergers, and that many of these will have measurable eccentricities in the LISA and LIGO detectors. However, what not yet have been quantified is the effect from the many more distant weak interactions the BBHs undergo in between their strong interactions. Weak interactions change the eccentricity of the BBHs, and can will therefore impact both their GW inspiral time and merger probability. We here use a simple cluster model to explore how BBHs are driven to merger under the influence of both weak- and strong interactions, as well as a GW emission. We find that including weak interactions leads to a notable increase in the number of BBHs that merge inside their cluster, which correspondingly leads to a higher number of eccentric LISA sources. These preliminary results illustrate the importance of including weak interactions for accurately modeling how BBHs merge in clusters, and how to link their emitted GW signals to their astrophysical environment.