It turns out that this norm can be computed efficiently (i.e., it is in $P$). This wasn't known at the time that the Davidson paper (originally linked in a comment above) was written, which is why it suggests that the computation is hard.
To compute the norm, first use the equivalence of points (1) and (2) in Theorem 1.1 of the Davidson paper. This tells us that $\|T\|_{schur} = \|T\|_{cb}$, where $\|\cdot\|_{cb}$ refers to the "completely bounded" norm of $T$.
It was shown in "J. Watrous. Semidefinite programs for completely bounded norms. Theory of Computing, 5:217-238, 2009" that the completely bounded norm can be computed efficiently via semidefinite programming.
For what it's worth, using this method tells me that the norm $\|\cdot\|_{schur}$ of the map stated in the question is $4$.