EDIT: I believe that I have come to the conclusion that my original idea was fundamentally misguided, and there is no reason to expect that such a process is possible in general. There is much more straightforward way to resolve the issue I wanted, which I have linked in my updated answer on the Physics site here.
My question is whether the existence of a generalized uncertainty relationship arising from the non-commutativity of Fourier transform duals leads to a chain rule restricted to either the variables or their duals: in my particular application this takes the form, $$ \frac{d\hat{A}}{dt} = \frac{\partial \hat{A}}{\partial t} + \sum_k \frac{\partial \hat{A}}{\partial x_k} \frac{\partial x_k}{\partial t} $$ or $$ \frac{d\hat{A}}{dt} = \frac{\partial \hat{A}}{\partial t} + \sum_k \frac{\partial \hat{A}}{\partial p_k} \frac{\partial p_k}{\partial t} $$
where $x_j$ are the spatial variables, $p_k$ are the momenta, and $\hat{A}$ is an arbitrary operator. In other words, does it necessarily follow that I can only write chain rule expansions in terms of the positions and momenta separately due to $[\hat{x},\hat{p}_x] = i\hbar$, which leads to the Heisenberg uncertainty principle in quantum mechanics, where $\hat{p}_x = -i\hbar \nabla$ in the $x$ representation and $\hbar$ is a real constant.
Additional context is that this question arose due to my inability to rationalize the restricted chain rule expansion used in Victor Stenger's ArXiv paper in equation 4.26, which also gives rise to the momentum space version of the operator equation that I have mentioned previously.
I will apologize in advance that I am likely not framing my question as technically as it could be since I am a physical chemist, not a mathematician. But I think that my question is fundamentally mathematical but just came up in the context of quantum mechanics.
Best Answer
This might be wrong, but I still think that the issue here is with proper function definition and does not tie directly with quantum mechanics.
Let's fall back to classical mechanics and let $A : \Gamma \times \mathbb{R} \rightarrow \mathbb{R}$ be some some observable, where $\Gamma$ is the phase space for some system with finite degrees of freedom, $q_i$, and we allow for time dependence.
You follow your system around while it evolves, and you may ask how does it change in time during that particular evolution path. Then, your question is about the derivative (technically of the composition of the observable with said path, but whatever) of $A$ with respect to $t$. Of course the chain rule will give you $$\frac{dA}{dt}=\frac{\partial A}{\partial t}+\sum_k \frac{\partial A}{\partial q_k}\dot{q}_k$$ Now does it make sense to make a "momentum expansion" of this observable? No; it isn't a function of the momenta. If it were, we would have to add $$\sum_k \frac{\partial A}{\partial p_k}\dot{p}_k$$ to the above expression right?
Now, if there were a functional dependence between the degrees of freedom and their momenta, then you would able to switch using the chain rule as well.
Then we come back to quantum mechanics. I'm not entirely sure what it means for an operator to depend on position or on momentum. I don't know if this would be a map $\hat{A}: \Gamma \rightarrow \mathbb{O}(H)$ where $\mathbb{O}(H)$ is the set of self-adjoint operators on $H$, or if, for example, you might be expanding the operator in some basis and then talking about the dependence of the component functions on the phase space. But either way, I think it's safe to say that it is impossible for $\hat{A}$ to depend both on the position and on the momenta, because that would mean knowing both with absolute certainty, and we of course cannot. By impossible, I don't mean mathematically; surely you can define that in a mathematically meaningful way, but, physically, what sense does it make? I am free to define it, but then I hand it to you and you say "I can't calculate this, I can't possibly have the data to calculate this map" and you'd be right.
In short, I might be wrong, and I hope someone corrects me if I am, but I don't think the problem is mathematical, apart from the technicality of defining precisely the function in question.