We investigate the dynamics of Bose-Einstein condensate droplets composed of ¹⁶⁴Dy atoms formed in a double-well potential following removal of the interwell barrier. By solving the dipolar Gross-Pitaevskii equation, we determine phase diagrams of ground-state configurations as functions of the atom number confined in the double-well potential. For an interwell separation 2d=3 μm, we consider the relative dipolar-to-contact interaction strengths ε_dd=1.4, 1.45, and 1.5. Symmetry-broken lowest-energy configurations are found for ε_dd=1.5 and, over a reduced range of atom numbers, for ε_dd=1.45, whereas no symmetry-broken states are found for ε_dd=1.4 up to N=5×10⁴. We analyze the subsequent time evolution after removal of the central barrier, revealing both droplet oscillations and merger events leading to the formation of larger droplets. The oscillations are driven by the external potential and by the repulsive tails of the in-plane component of the dipolar interaction. For the two-droplet states at ε_dd=1.5 and 2d=3 μm, merging is found for N=8000 and 8800, whereas the droplets remain separated for the investigated cases with N≥9000. Increasing the initial separation to 2d=5 μm shifts the approximate upper atom number for merging to N≃1.65×10⁴, demonstrating that the crossover is controlled by the competition between the post-quench excess energy and the repulsive inter-droplet barrier.