Let's admit that the vibration energy is not counted by half as kinetic energy, but separately.
"The temperature is not changing" is no good reason. Molecules have an individual speed and kinetic energy in the liquid, they don't in the solid. Here the kinetic energy changes at constant temperature.
But if we recognized that vibration is just half kinetic energy and half deformation energy, the situation would be much more complicated and about impossible to answer. In stiff small liquid or gaseous molecules at moderate temperature, some vibration modes are not excited and store no energy, while if building a solid, they have collectively more vibration modes at lower energies that are excited and store energy. That's why nearly all metals have 3R/mol heat capacity. Then freezing may increase the energy stored in vibrations, and in some molecules, this increase could exceed the loss of energy in individual translations.
About potential energy: intermolecular forces and energies change in a liquid when, for instance, the liquid expands. No melting needed. You could compare the heat capacity of the same substance at the same temperature, as a liquid and as a gas. Big difference. Tables for water and steam are easily found.