Thanks for your PS!

@nflanders I think my example, when normalised, fits your setup, with $b(x,y)= n^{-1/2}\sum_{k=1}^n x_k y_k$ so that $|b(x,y)|\le \|x\|_2\|y\|_\infty$ (i.e., $C=1$). So one should replace $T=T_n$ above with $T_n/\sqrt{n}$. -- I think you have misread Prop.\ 1.5.3 that does not say what you quote, but that $\pi_2(T)$ equals the (quasi-) norm of $T$ in the product operator ideal $\mathfrak{H}\circ \mathfrak{P}_2$, cf.\ ibidem D.1.10.

I think what you are asking is this: If $H$ is a Hilbert space, $X$ ($=B^*$) is a Banach space, $T:H\to X$ is an operator, $i: X\to H$ is a continuous injection, can one estimate the 2-summing norm $\pi_2(T)$ by a multiple of $\pi_2(iT)$? Take $H=\ell_2$, $X=\ell_1$, $i=$ the identity mapping and $T=T_n=$ the projection onto the first $n$ coordinates. Then $\pi_2(iT_n)= \sqrt{n}$, but $\pi_2(T_n)$ is of order $n$; see 1.6.8 in Pietsch's Eigenvalues and $s$-Numbers.

Returning to your question concerning reflexive spaces: The dual of $V\otimes_\varepsilon W$ is then $V^* \otimes _\pi W^*$ provided one of these spaces has Grothendieck's approximation property. (There's a more general such theorem involving spaces with the Radon-Nikodym property.)