In the case of $\text{GBC}^-$, the answer is no, $\Diamond$ is not provable from the assumption that all reals are constructible. It is consistent with $\text{GBC}^-$ and even $\text{KM}^-$ that $\Diamond$ fails, yet every real is constructible.
To produce a model of this, start in $L$. Evidently there is a way to force Suslin's Hypothesis (killing all Suslin trees) without adding reals. My understanding is that Jensen provided such an argument in the 1970s, but there are better methods available for this today. (See the answers on this thread.) Let me black box that argument and proceed.
If $L[G]$ is the resulting extension, then every real is constructible, yet $\Diamond$ fails. This will be visible in the $\text{KM}^-$ model arising from the hereditarily countable sets $H_{\omega_1}$ and it classes in $L[G]$.
The relevant feature of the model is that knowing every real is constructible does not imply that every class is constructible, since in effect the forcing created new classes over $H_{\omega_1}$, but no new sets.
A very similar phenomenon arises with KM class theory, where the assertion $V=L$ can be seen as ambiguous. Namely, knowing that $V=\bigcup_\alpha L_\alpha$ does not imply that every class arises constructibly in the continued iteration of the $L$-hierarchy beyond Ord using class codes for metaordinals. This issue is connected with the Class Collection (aka Class Choice) principle. The CC axioms does not hold in all KM models, but it does hold in models of V=L, provided also that every class is constructible. And furthermore, one can cut down the KM model to include just those classes that are constructible, and thereby see that KM and KM+CC are mutually interpretable.