I can't really think of any other way of calculating the lift
coefficient of an airfoil theoretically.

A lot of good people have found more straightforward ways to compute the
lift coefficient, which is in fact very easy to calculate. The maximum
lift coefficient, the drag coefficient, the pitching moment coefficient
are the hard ones.

Perhaps you could consult a standard textbook on glider aerodynamics, e.g.
Prof. Fred Thomas "Handbook of Sailplane Design".