= Solution
The <Hadamard basis> changes the <Pauli matrices> by $Z_S=\widetilde X_S$ and $X_D=\widetilde Z_D$. Thus, keeping tensor factors ordered as $S,D$, the <Hamiltonian operator> is
$$
\boxed{H_{\rm int}(t)=\frac{\pi g(t)}4
(I_S-\widetilde X_S)\otimes(I_D-\widetilde Z_D).}
$$
If $D$ has new-basis label plus, the second factor vanishes. If $D$ has label minus, the integrated generator on $S$ is $\pi(I-\widetilde X_S)/2$, whose exponential is $\widetilde X_S$ by part (a). The generator therefore has the same controlled form with $D$ as control in the complementary bases.
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