Embedded in an ohmic environment, the Josephson current peak can transfer part of its weight to finite voltage and the junction becomes resistive. The dissipative environment can even suppress the superconducting effect of the junction via a quantum phase transition occurring when the ohmic resistance $R_s$ exceeds the quantum resistance $R{}_{\text{q}}=h/{\left(2e\right)}^2$. For a topological junction hosting Majorana bound states with a $4\pi$ periodicity of the superconducting phase, the superconductor-insulator phase transition is shifted to $4R{}_{\text{q}}$. We consider a Josephson junction mixing the $2\pi$ and $4\pi$ periodicities shunted by a resistor, with a resistance between $R_q$ and $4R_q$ such that the two periodicities promote competing phases. Starting with a quantum circuit model, we derive the nonmonotonic temperature dependence of its differential resistance resulting from the competition between the two periodicities, the $4\pi$ periodicity dominating at the lowest temperatures. The nonmonotonic behavior is first revealed by straightforward perturbation theory and then substantiated by a fermionization to exactly solvable models when $R_s=2R{}_{\text{q}}$: the model is mapped onto a helical wire coupled to a topological superconductor when the Josephson energy is small and to the Emery-Kivelson line of the two-channel Kondo model in the opposite case. We also settle the compact vs extended phase controversy: introducing a compact phase variable across the junction, associated with a discrete charge, we rigorously prove that it is effectively replaced by an extended phase variable in presence of the environment.