Fluxons in a superconducting loop can be coherently coupled by quantum phase slips occurring at a weak link such as a Josephson junction. If Cooper pair tunneling at the junction occurs through a resonant level, then 2π quantum phase slips are suppressed, and fluxons are predominantly coupled by 4π quantum phase slips. We analyze this scenario by computing the coupling between fluxons as the level is brought into resonance with the superconducting condensate. The results indicate that the 4π-dominated regime can be observed directly in the transition spectrum for circuit parameters typical of a fluxonium qubit. We also show that if the inductive energy of the loop is much smaller than the plasma frequency of the junction, then the low-energy Hamiltonian of the circuit is dual to that of a topological superconducting island. These findings can inform experiments on bifluxon qubits as well as the design of novel types of protected qubits.