It seems to me that ultimately you do not care about the numerical error in the finite-difference weights themselves, but in the numerical error you incur when you use these weights. Assessing the latter error is very simple. Just plug into your approximation constant function values. Assuming your weights approximate a derivative, you should obtain identically zero. Most likely you will not obtain exactly zero, but a value on the order of the machine precision. If your result is much larger than that, the cause might be numerical errors in the computed weights.
You can carry out this test also for higher-order functions. Assuming you have a second-order accurate approximation of the second derivative, you will have a TE that is proportional to
Now the fourth derivative of a cubic polynomial is zero, so your approximation should give you the exact second derivative if you plug in the function values of a cubic polynomial. The deviation between the exact derivative and your computed derivative is again an indication of the numerical error in your weights.
In my use of Fornberg's algorithm, I did not see any indication that errors in the numerically computed weights were significant.