We study n-pulse interaction in fast collisions of N solitons of the cubic nonlinear Schrödinger (NLS) equation in the presence of generic weak nonlinear loss. We develop a generalized reduced model that yields the contribution of the n-pulse interaction to the amplitude shift for collisions in the presence of weak (2m+1)-order loss, for any n and m. We first employ the reduced model and numerical solution of the perturbed NLS equation to analyze soliton collisions in the presence of septic loss (m=3). Our calculations show that the three-pulse interaction gives the dominant contribution to the collision-induced amplitude shift already in a full-overlap four-soliton collision, and that the amplitude shift strongly depends on the initial soliton positions. We then extend these results for a generic weak nonlinear loss of the form G(|ψ|2)ψ, where ψ is the physical field and G is a Taylor polynomial of degree mc. Considering mc=3, as an example, we show that three-pulse interaction gives the dominant contribution to the amplitude shift in a six-soliton collision, despite the presence of low-order loss. Our study quantitatively demonstrates that n-pulse interaction with high n values plays a key role in fast collisions of NLS solitons in the presence of generic nonlinear loss. Moreover, the scalings of n-pulse interaction effects with n and m and the strong dependence on initial soliton positions lead to complex collision dynamics, which is very different from that observed in fast NLS soliton collisions in the presence of cubic loss.