We introduce a complete model for dephasing due to line-of-sight acceleration (LOSA) in gravitational wave (GW) signals from stellar-mass binary black holes (BBHs) in three-body systems. Our prescription provides curvature- and projection-dependent phase features that are not recovered by local-expansion-based treatments. We perform parameter-space surveys and mock parameter inferences assuming the nominal sensitivity of the Einstein Telescope (ET) to identify the regime where the time-varying LOSA allows for separate constraints on the outer orbital parameters, in particular the tertiary mass and distance. We estimate that ET may detect a few to tens of such systems per year, provided that all binaries merge dynamically, and demonstrate that these constraints can be used to directly discriminate between a dynamical and AGN origin for BBHs. Finally, we reanalyse the GW190814 event and four O4a events finding no evidence for LOSA, with the previously claimed LOSA in GW190814 disappearing when a sufficiently long data segment is used.