Convert NURBS curve into Cubic Bezier Curve
I assume that your cubic spline is non-rational, meaning $w$ = 1, then this is in general true that exact conversion is not possible. NURBS are rational, so you can model conics, say circle whereas Bézier cannot. There is special case: rational Bézier curve, which could be used in place of NURBS.

This simply means that you can draw circle, and only approximate it with Bézier.

To approximate, it depends on degree of curve: least squares method is stable and fast for low degrees, but fails (error is not equal along curve and huge) for higher degrees. Another idea is like in de Casteljau, recursively subdivide curve and use piecewise approximation. This idea generates lots of points and is numerically less stable than least squares.

For general NURBS of order 6+, approximation technique must be of order at least 6+, but it converges slowly.

In any case, finding Bézier curve through points from NURBS is way to go.

Another option is lossless conversion from NURBS to piecewise Rational B-Splines (URBS) (after conversion these are uniform, but no precision is lost), then via Boehm algorithm or Greville points convert B-spline to R-Bézier (here again, nothing is lost) and then project it to NR-Bézier (approximation, i.e. https://ieeexplore.ieee.org/document/6658158 or https://arxiv.org/pdf/1212.3385.pdf, here conics are lost).

Errors are seen about five-six places after decimal point for nasty splines (these that cannot be exactly recovered, as oposed to nice, which are wxact), so for any drawing purpose it is not visible, unless this is for milling machine, but hey, CNC supports (G-code G02) arcs anyway.