I've come to the conclusion that it's not worth doing for my application, so I thought I would post my progress here for anyone else who might be interested in this same problem.
My more "proper" solution, which does not put any constraints on the differential equation but instead uses an error estimation for the adaptive steps, more like a ...
I followed D.W.'s approach and found a solution as follows:
In order to use an online derivative calculator, I expressed the cubic spline with points P0, P1, P2 and P3 using variables a through h:
P0 = (a, e)
P1 = (b, f)
P2 = (c, g)
P3 = (d, h)
The x-y-coordinates and their derivatives are as follows:
x(t) = a*(1-t)^3+3*b*(1-t)^2*t+3*c*(1-t)*t^2+d*t^3