  We numerically compute the low-energy spectrum of a gate-controlled
superconducting topological nanowire segmented into two islands, each
Josephson-coupled to a bulk superconductor. This device may host two pairs of
Majorana bound states and could provide a platform for testing Majorana fusion
rules. We analyze the crossover between (i) a charge-dominated regime
utilizable for initialization and readout of Majorana bound states, (ii) a
single-island regime for dominating inter-island Majorana coupling, (iii) a
Josephson-plasmon regime for large coupling to the bulk superconductors, and
(iv) a regime of four Majorana bound states allowing for topologically
protected Majorana manipulations. From the energy spectrum, we derive
conservative estimates for the time scales of a fusion-rule testing protocol
proposed recently [arXiv:1511.05153]. We also analyze the steps needed for
basic Majorana braiding operations in branched nanowire structures.
