= Solution
For a block-diagonal element, form preservation says
$$
ARC^T=R,\qquad BJ_UB^T=J_U.
$$
Given any $C\in\operatorname{GL}(W)$, the first equation uniquely determines
$$
\boxed{A=RC^{-T}R.}
$$
With dual bases ordered in matching rather than reversed order, the same relation is simply $A=C^{-T}$. This is the contragredient action on the paired space $W'$.
The second equation independently allows every $B\in\operatorname{Sp}(U)$. The map taking a block-diagonal element to $(C,B)$ is a group isomorphism, with inverse $ (C,B)\mapsto(RC^{-T}R,B,C,0,0,0)$. Thus
$$
\boxed{L\cong\operatorname{GL}_k(q)\times\operatorname{Sp}_{2m-2k}(q).}
$$
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