How to calculate the lift coefficient of a multi element airfoil?
It depends. Are the c$_l$ values referenced to the local chord? Then you need to convert them to the chord of the full airfoil so they can be added.

I wonder, however, how you can have individual lift coefficients without the total lift coefficient. If the single parts are isolated, their pressure distribution is different from that in the context of the full airfoil. Please check the source of the lift coefficients and make sure that they are representative of the local lift when flown in combination with all other parts.

Normally, what needs to be done is to use a multi-element airfoil code and calculate the lift coefficient of the full wing with all high-lift devices in place.