The molecular Monte Carlo method, combined with the CRG interatomic potential, is used for the first time to investigate stoichiometric mixed oxides U$_{1-y}$Pu$_y$O$_2$ (with y in the range 0–1). The implementation of this Monte Carlo method for mixed oxides simulation was carried out involving two algorithms, with and without cation exchange. The use of these two Monte Carlo algorithms allowed us to test the effect of the substitutional disorder implied by the coexistence of two types of cations. Structural, thermodynamic, and mechanical properties of the stoichiometric mixed oxides fuel U$_{1-y}$Pu$_y$O$_2$ have been investigated over a wide temperature range (from 300 K to the melting temperature) and plutonium content (from 0 to 100 atom %). Our study shows that the exploration of cationic configurations through the cation exchange algorithm is required for a complete description of the mixed oxides fuel properties, especially for the atomic structural properties. Concerning thermodynamic properties, the evolution of the computed specific heat as a function of temperature exhibits one peak for all plutonium contents around 2300 K, i.e. at $\sim$0.8 T$_m$ (T$_m$ is the melting temperature). The same behavior is observed for the linear thermal expansion coefficient. These peaks, also observed in previous studies, are related to the Bredig transition known to occur around 0.8 T$_m$. A good agreement between our results, experiments, and previous calculations is found for temperatures up to about 2100 K. Above this temperature, our calculations show a behavior different from experimental recommendations.