Previous answers have pointed out why you can expect an exponential behaviour of the specific heat when the system's spectrum is gapped, thus explaining the superconductor's specific heat at small temperatures.
But if I am not wrong, you are confused by the fact that $C_v(T\to T_c^-) > C_v(T\to T_c^+)$ (where $C_v$ is the specific heat and $T_c$ the critical temperature). In words, the specific heat of the superconducting state close to the critical temperature is higher than the specific heat of the normal metallic state close to the critical temperature.
You said:
If we have an energy bandgap, shouldn't we need more energy to heat up
such a system than to heat up a "normally conductive" system with
continuous energy?
but I think this is not the correct interpretation. $C_v = d\left\langle E \right\rangle /dT$ measures the slope of the internal energy vs temperature curve. So what you said is correct: the internal energy in the metallic state is higher than in the superconducting state because the spectral gap is closed, but the rapidity of energy growth with temperature is higher for the superconducting state. When $T \to T_c^-$, the internal energy grows faster with $T$ than in the case $T \to T_c^+$, even though the energy always grows with temperature.
The physical meaning is clear: as $T\to T_c^-$, the energy spectrum changes: the gap $\Delta(T)$ closes very quickly, roughly $\Delta(T) \sim k_BT_c\sqrt{ 1 - \frac{T}{T_c}}$, so the excitation of quasiparticles is easier and easier as $T$ increases. When $T>T_c$ the gap is closed and the spectrum does not change with $T$, thus the energy gain is slower.