Apter and I proved that the only way to force this situation by
any forcing that resembles anything like the Laver preparation is
to begin with a supercompact cardinal (or a partially supercompact
cardinal for $\theta$ fixed). Specifically, in our article A. W.
Apter and J. D. Hamkins, “Indestructible weakly compact cardinals
and the necessity of supercompactness for certain proof schemata,”
MLQ Math. Log. Q., vol. 47, iss. 4, pp. 563-571,
2001, we prove:

**Theorem 2.** If after forcing with a closure point below
$\kappa$ the weak compactness of $\kappa$ becomes indestructible
by the forcing to collapse cardinals to $\kappa$, then $\kappa$
was supercompact in the ground model.

**Theorem 3.** Suppose $\kappa$ is weakly compact in a forcing
extension that admits a closure point below $\kappa$ and that
collapses $(2^{\theta^{<\kappa}})^V$ to $\kappa$. Then $\kappa$ was
$\theta$-supercompact in $V$.

The proof really uses only that the universe has the
$\delta$-approximation and cover properties over the ground model,
for some $\delta<\kappa$.

This result does not settle the exact consistency strength,
because of the assumption on the forcing, but provides fairly
strong evidence that a supercompact cardinal is required for
arbitrary collapses and a $\theta$-supercompact is required for
collapsing $2^{\theta^{<\kappa}}$.

Meanwhile, as in your remark 2, I am given to understand from the inner model theory
experts that having a weakly compact cardinal that is
indestructible by collapse forcing to $\kappa$ outstrips the inner
model theory, going past Woodin cardinals and more. But we'll have
to hear the details from the inner model theory experts.