Given a language $\mathcal L$ and an $\mathcal L$-structure $\mathcal M$, the theory of $\text{Th}(\mathcal M)$ is defined to be the set of $\mathcal L$-sentences $\phi$ for which $\mathcal M \vDash \phi$. Such theories are maximal, meaning that for each $\mathcal L$-sentence $\phi$ either $\phi \in \text{Th}(\mathcal M)$ or $\neg \phi \in \text{Th}(\mathcal M)$. This is because either $\mathcal M \vDash \phi$ or $\mathcal M \nvDash \phi$, simply by merit of the interpretation function $\cdot^\mathcal M : \text{Formula}(\mathcal L) \to \{T,F\}$, uh, existing.
Although theories of $\mathcal L$-structures "are maximal", this fact may not be realizable. By this I mean the following: there may exist an $\mathcal L$-sentence $\phi$ for which, despite that $\phi \in \text{Th}(\mathcal M) \lor \neg \phi \in \text{Th}(\mathcal M)$, it is the case that $\phi \in \text{Th}(\mathcal M)$ is not provable and $\neg \phi \in \text{Th}(\mathcal M)$ also is not provable.
Let me give a specific example. For this example I assume that $\Phi$ is some undecidable sentence within which the system we are currently working; for instance, assuming we are working in ZFC (within which we have encoded model theory), we could have $\Phi$ be $2^{\aleph_0} = \aleph_1$.
Let $\mathcal L = \{S, z\}$ be the language of the natural numbers, and $\Gamma$ the axioms of Robinson arithmetic. Define a model $\mathcal M$ for $\Gamma$ as follows: if $\Phi$ is false, then let $\mathcal M = (\mathbb N;\ 0, +)$ be the standard model of the naturals; if $\Phi$ is true, then let $\mathcal M = (\mathbb N \cup \{\infty\};\ 0, +)$ be the standard model plus an additional element $\infty$ for which $S(\infty) = \infty$.
Note that in this example we still have that $\text{Th}(\mathcal M)$ "is complete", but not in any real way. Letting $\phi$ be the sentence $\exists i : S(i) = i$, we have since $\mathcal M$ is well-defined that either $\mathcal M \vDash \phi$ or $\mathcal M \vDash \neg \phi$, but in fact it is impossible to prove either way due to undecidability of $\Phi$.
My questions are these:
Are there any examples of this effect that are less "gimmicky"? In my example I basically shoehorn in a poorly-behaved $\phi$ by assuming existence of an undecidable $\Phi$. Is there an example where the $\phi$ arises "naturally" from the model, not requiring existence of an external undecidable $\Phi$?
Are there any ((meta-)meta-)theorems along these lines? For instance, here's a conjecture: if $\Gamma$ is strong enough for Goedel's incompleteness theorem to apply, and if $\mathcal M \vDash \Gamma$, then there exists a sentence $\phi$ for which neither $\mathcal M \vDash \phi$ nor $\mathcal M \nvDash \phi$ are provable, despite their disjunction being true.
(I have a small handful of other examples of this effect, but they are similarly gimmicky. I also tried my hand at proving this conjecture, but eventually concluded that my method is likely flawed.)
As often happens with logic, there are some subtleties around precisely posing the question you intend to ask. Let me rephrase it as follows:
Fix a "background theory" to do model theory in - say, $\mathsf{ZFC}$. A definition of a structure will be a formula $\varphi$ in the language of set theory such that $\mathsf{ZFC}\vdash$ "$\varphi$ defines a structure" (note that part of this is that $\mathsf{ZFC}\vdash\exists! x\varphi(x)$). When $\varphi$ is such a formula, I'll write "$\mathcal{M}_\varphi$" for the structure defined by $\varphi$. Note that different background models of $\mathsf{ZFC}$ may compute $\mathcal{M}_\varphi$ differently, but for any fixed $W\models\mathsf{ZFC}$ there will be a unique-and-well-defined value of $\mathcal{M}_\varphi^W$ ("what $W$ thinks $\varphi$ defines"). Similarly, let $\Sigma_\varphi$ be the language of $\mathcal{M}_\varphi$; for simplicity, let's only look at situations where $\Sigma_\varphi$ is finite and "fully pinned down" by the $\mathsf{ZFC}$ axioms (this just makes things somewhat easier to think about).
Now say that a definition of a structure $\varphi$ is bivalent iff for every $\Sigma_\varphi$-sentence $\theta$ we have $$\mathsf{ZFC}\vdash\theta^{\mathcal{M}_\varphi}\quad\mbox{or}\quad\mathsf{ZFC}\vdash\neg\theta^{\mathcal{M}_\varphi}.$$ You're essentially asking about non-bivalent definitions of structures.
One quick observation is that, as you suspect, any $\varphi$ which $\mathsf{ZFC}$-consistently defines a structure rich enough to implement arithmetic must be non-bivalent. This is because we can encode the Godel-Rosser sentence for $\mathsf{ZFC}$, or indeed any c.e. extension of $\mathsf{ZFC}$, as a $\Sigma_\varphi$-sentence. In particular, if we let $\varphi$ be the usual definition of the natural numbers with addition and multiplication inside set theory, then $\varphi$ is non-bivalent. Indeed, Godel's incompleteness theorem shows that this remains the case no matter how we strengthen $\mathsf{ZFC}$, as long as we remain consistent and c.e.
On the other hand, bivalent definitions of structures also exist! The usual definition of the field of real numbers is an example; see this old answer of mine.
Now on to the topic of gimmicks. Informally, you've shown that every structure which has a bivalent definition also has a non-bivalent definition; a bit more precisely, if $\varphi$ is a bivalent definition of a structure, then letting $\theta$ be a sentence independent of $\mathsf{ZFC}$ there are non-bivalent definitions of structures $\varphi_1,\varphi_2$ such that $\mathsf{ZFC}$ proves that $\varphi_1$ coincides with $\varphi$ iff $\theta$ and $\varphi_2$ coincides with $\varphi$ iff $\neg\theta$. This raises the natural question of whether there is a structure which has "copies" definable in every model $\mathsf{ZFC}$ but has no bivalent definition:
I believe the answer to this is no, and I have a sketch of an argument, but I don't have time to flesh it out right now; I'll add it later today if it holds up, and if it doesn't hold up I'll explain why it breaks.