We study phase-controlled transport through a one-dimensional channel of attractively interacting fermions connecting two three-dimensional BCS reservoirs. The competition between channel phase stiffness and pair tunnelling at the two contacts results in the emergence of two 4π-periodic families of stationary states, characterised by symmetric or asymmetric phase drops at the boundaries. Advancing the reservoir phase difference by 2π exchanges these families so that the equilibrium energy remains 2π periodic, while the 4π periodicity might be attained in a diabatic regime. This structure admits a formal description in terms of two boundary Majorana operators, combining into a nonlocal fermion mode. Renormalization-group analysis shows that quantum fluctuations and phase slips are suppressed in the zero-temperature thermodynamic limit, preserving the branch crossings. The predicted branch exchange, metastable currents, and sweep-rate-dependent switching provide experimentally testable signatures of Majorana-like Josephson branch physics in a gapless interacting channel.