We study the escape time and its relationship with phase space structures that control the transport in a Hamiltonian model for an exothermic chemical reaction for different energies. The classical Hamiltonian model is a variant of a model proposed to study very exothermic reactions in ultracold systems. The original model considered two elements to determine the evolution of the system: a van der Waals potential energy and random kicks to simulate the interaction with the other particles. The present model uses random bumps in the potential energy to simulate the effects of the other particles in the system. We compare the results of both models and explain their differences and similarities. In order to study the phase space of the system, we use the method of Lagrangian descriptors to detect phase space structures relevant to the dynamics like stable and unstable manifolds of the hyperbolic orbits, the KAM islands and sticky regions around them. The Lagrangian descriptors are useful tools to reveal the structure of the invariant manifolds that direct the dynamics in the phase space in systems where the standard tools require multiprecision calculations.