If you want to prevent cracks between the tessellated lines in a single curve, what you need to know is how to miter the quads correctly. Your GS's extrusion is presumably based on taking the 2D perpendicular to the line direction and extending the quad in that direction. Well, don't do that.
Instead, have the TES compute the perpendicular direction for each vertex in the tessellation. You can compute this by taking the first derivative of the Bezier curve (its tangent), which can be computed exactly, and then finding the perpendicular of that. Each vertex will have its own perpendicular value.
It should be this perpendicular which you feed to the GS to do the quad extrusion. And you should be guaranteed to have invariant results. The TES's computation of the position and perpendicular will be based solely on the positions of the control points and the tessellation coordinates. The control points are fixed for all invocations of a TES over a patch, and the tessellator guarantees the invariance of
gl_TessCoord for the "same vertex".
Invariance between patches will be something of a problem. That is, invariance between the end of one set of 4 control points and the beginning of an attached one. The tessellation system only guarantees invariance at edges between patches if the computations that lead to the positions of the vertices are all based solely on data that is binary identical between the two edge vertices. And that's not generally the case for bezier curves; you achieve C1 continuity by ensuring that the slope between boundary control points is the same. And that slope is not passed as a parameter; it is computed. And thus, two edge vertices on different patches could have slightly different slope computations, which leads to different vertices, which leads to invariance.
The only way to handle this is to have your Bezier spline defined by six control points rather than four. You pass as the first and last control points the previous/next control points for the Bezier spline (you'll have to make something up at the edges). This allows your computations to be invariant, since at the edges, you will be doing computations based on the exact same data.