In Variational "Design of Rational Bézier Curves and
Surfaces" Georges-Pierre Bonneau's PhD thesis it's stated that isocurves (which are curves obtained fixing one parameter of a parametrized surface) of a rational Bézier surface are rational Bézier curves. See remark 1.33 below.
It's easy to prove that for a non rational Bézier surface but i'm stuck in the rational case. The denominator $W(u,v)$ functions suggests me that, for a fixed $u_0$, the weights of the isocurve should be $\sum\limits_{i=0}^{m}B_{i}^{m}w_{i,j}$ as we have $$W(u_0,v)=\sum\limits_{j=0}^{n}\sum\limits_{i=0}^{m}B_{j}^{n}(v)B_{i}^{m}(u_0)w_{i,j}=\sum\limits_{j=0}^{n}B_{j}^{n}(v)\left( \sum\limits_{i=0}^{m}B_{i}^{m}(u_0)w_{i,j}\right)$$ but that doesn't seem to work with the numerator. I'm not able to extract a formula with such weights (the main problem is that both $w_{i,j}$ and $\mathbf{b}_{i,j}$ depend on both indices).
Link to the thesis: https://tel.archives-ouvertes.fr/tel-01064604/document