This paper is devoted to the construction of a discontinuous Galerkin discretisation (DG) for the nonconservative bitemperature Euler system via a a discrete BGK formulation. This formulation is compatible with the entropy properties of the system and thus provides admissible solutions. The DG method is used to approximate the linear transport part of the BGK model while the force and source-terms are treated implicitly but with explicit expressions. High order in time has also been investigated using SSP Runge-Kutta methods. We numerically show the good agreement of our results with the ones provided by other schemes, including solutions with shocks.