I need this rope to continue wrapping after 250 frames - and be able to render properly.
Here's a solution:
Raise the value of the End frame (to increase the duration of the animation). The default value is 250. Raise it to e.g. 500.
Increase the duration of rotation of Cylinder. Select object 'Cylinder'. Change the current view to Graph Editor. Press [Home] to center the view. Press [A] in the Graph Editor to select all vertices. In the menu of the Graph Editor go to [Channel]-[Extrapolation Mode] and select [Lineair Extrapolation]. The curves will be alongated.
Raise the cache of the rope. Select the rope (Nurbspath) and move to the Physics properties in the properties window. Within the section [Cache] raise the value [End] to a higher value (e.g. 500).
Move to the first frame and run the animation.