Fitting data to a constant multiplied by a reference spline

I have a data that I need to fit to a spline function multiplied by some constant.

I fit a reference function (spline) using: refSpline=fit(xData,referenceData,'smoothingspline');

Now, I would like to use refSpline multiplied by some weighting factor (c) to fit a data set of the same dimensions. The function is not parametric (as far as I understand it), so can this curve even be used for fitting? I could use refSpline to output a Y vector, and do a chi-squared optimization, but I would like to avoid that, as I have no experience with writing such optimizations. Any help is appreciated.

Best, Michael