As the voltage between the capacitor's plates decreases, so should the current flowing through the circuit.
I don't follow your reasoning here. Recall that, for an ideal capacitor, we have:
$$i_C = C\frac{dv_C}{dt}$$
In words, the current through the capacitor is proportional to the rate of change of the voltage across, not the instantaneous value of the voltage.
So, for example, if the voltage across the capacitor is sinusoidal
$$v_C = V \sin\omega t$$
the current is
$$i_C = \omega V \cos \omega t$$$$i_C = \omega CV \cos \omega t$$
which means (1) that the maximum current (magnitude) occurs when the voltage is zero and (2) that the maximum voltage (magnitude) occurs when the current is zero.
Now, for this simple LC circuit, the voltage across the capacitor is identical to the voltage across the inductor:
$$v_C = v_L$$
thus,
$$i_C = C\frac{dv_L}{dt}$$
For an ideal inductor, we have:
$$v_L = L\frac{di_L}{dt}$$
But, the inductor current is
$$i_L = - i_C$$
thus,
$$i_C = -LC\dfrac{d^2i_C}{dt^2}$$
which means that the current is sinusoidal
$$i_C = A \sin \omega t + B \cos \omega t $$
where
$$\omega = \frac{1}{\sqrt{LC}}$$
Since, in your example, the initial current is zero and the initial voltage is $V$, we have
$$i_C(t) = -\frac{V}{\omega L} \sin \omega t$$