We study a class of interacting particle systems on $\mathbb{R}$ which was recently investigated by F. Götze and the second author [GV14]. These ensembles generalize eigenvalue ensembles of Hermitian random matrices by allowing different interactions between particles. Although these ensembles are not known to be determinantal one can use the stochastic linearization method of [GV14] to represent them as averages of determinantal ones. Our results describe the transition between universal behavior in the regime of the Tracy-Widom law and non-universal behavior for large deviations of the rightmost particle. Moreover, a detailed analysis of the transition that occurs in the regime of moderate deviations, is provided. We also compare our results with the corresponding ones obtained recently for determinantal ensembles [Sch15, EKS15]. In particular, we discuss how the averaging effects the leading order behavior in the regime of large deviations. In the analyis of the averaging procedure we use detailed asymptotic information on the behavior of Christoffel-Darboux kernels that is uniform for perturbative families of weights. Such results have been provided by K. Schubert, K. Schüler and the authors in [KSSV14].