There’s a risk that I’ve missed some nuances of the discussion. But I don’t think a general optimization routine is needed to find optima of a cubic spline. You’re just dealing with a bunch of 3rd degree polynomials, each of which should have an easily accessible analytic optimum for each interval.
Calculating optima of splines should be a well-known and solved problem. If the spline/interpolation package doesn’t provide a built-in max-method, can’t you find the solution on Wikipedia?