How to design LC filter for smoothing out sine wave from sine PWM?
Normally, a good starting point for the LC filter is choosing the cut-off frequency and that can be found by finding the logarithmic half-point between 50 Hz and 15 kHz (in your example). Numerically it is \$\sqrt{50 \cdot 15000}\$ = 866 Hz. It's not a hard rule but something to get you started.

So this constrains L and C somewhat but loading effects can reduce the Q of the LC circuit and this is probably best done using a simulator like micro-cap 11 or LTSpice (both free). Just model the circuit with with highest and lowest loads and see what happens to the shape of the sine wave and its amplitude.

I'd consider starting with L = 10 mH and C = 3.3 uF (876 Hz) with a little series resistance with the inductor to avoid Q factor rising too high on low output currents.