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[Abridged] Binary black holes may form and merge dynamically. These binaries are likely to become bound with high eccentricities, resulting in a burst of gravitational radiation at their point of closest approach. When such a binary is perturbed by a third body, the evolution of the orbit is affected, and gravitational-wave burst times are altered. The bursts times therefore encode information about the tertiary. In order to extract this information, we require a prescription for the relationship between the tertiary properties and the gravitational-wave burst times. In this paper, we demonstrate a toy model for the burst times of a secular three-body system. We show how Bayesian inference can be employed to deduce the tertiary properties when the bursts are detected by next-generation ground-based gravitational-wave detectors. We study the bursts from an eccentric binary with a total mass of $60$~M$_\odot$ orbiting an $6 \times 10^{8}$~M$_\odot$ supermassive black hole. When we assume no knowledge of the eccentric binary, we are unable to tightly constrain the existence or properties of the tertiary, and we recover biased posterior probability distributions for the parameters of the eccentric binary. However, when the properties of the binary are already well-known -- as is likely if the late inspiral and merger are also detected -- we are able to more accurately infer the mass of the perturber, $m_3$, and its distance from the binary, $R$. When we assume measurement precision on the binary parameters consistent with expectations for next-generation gravitational-wave detectors, we can be greater than $90\%$ confident that the binary is perturbed. [...]