Setup
In field theory, one usually is given a Lagrangian volume form which depends on some fields (and finitely many of their derivatives) which are differential forms. The equations of motion of such system are obtained by performing infinitesimal perturbations of such fields. This is known as the first variational formula, and it can be stated generically as: given a Lagrangian D-form $\textbf{L}[\varphi]$ which depends on some fields $\varphi$ which are $p$-forms generically (and their derivatives), then under infinitesimal perturbations of the fields, $$\delta \textbf{L}[\varphi]=\frac{\delta \textbf{L}}{\delta \varphi} \wedge \delta \varphi+d \mathbf{\theta}(\varphi,\delta\varphi),$$ where $\frac{\delta \textbf{L}}{\delta \varphi}=0 $ are the equations of motion $(D-p)$-forms. This result relies on integration by parts and some properties of volume forms are typically required. Let me exemplify with an extremely simple example. Assume that we are given a Lagrangian D-form $\textbf{L}$ which depends on a one-form $A=A_\mu dx^\mu$ as $\textbf{L}[A]=\frac 12 A\wedge\star A$.
Perturbing the field infinitesimally $A\mapsto A+\delta A$, $$\textbf{L}[A+\delta A]=\frac 12 (A+\delta A)\wedge\star (A+\delta A)=\frac 12 A\wedge\star A+(-1)^{D-1} \star A\wedge \delta A+\frac 12 \delta A\wedge\star \delta A,$$
Disregarding the terms of order 2 in the perturbations,
$$\delta \textbf{L}[A]=\textbf{L}[A+\delta A]-\textbf{L}[A]=(-1)^{D-1} \star A\wedge \delta A$$ we indeed obtain that an arbitrary infinitesimal perturbation of the Lagrangian can be generically written as a term proportional to linear perturbations in the fields up to total derivatives.
The problem
So far there is no problem. The issue arises when I try to study perturbations of differential forms which are not volume forms. For instance, if I try to compute how the $(D-1)$-form $\star A$ transforms under perturbations of $A$ (the metric is kept fixed), I am unable to write the result in the same fashion as in the first variational formula $$\delta \star A\neq (\text{something})\wedge\delta A.$$
Thus, I was wondering if there is an analogue result to the first variational formula for non-volume forms, i.e. forms of degree less than $D$. The motivation is that, when deriving the equations of motion of a Lagrangian, one typically writes in a formal way
$$\delta \textbf{L}[A,dA]=\frac{\partial \textbf{L}}{\partial A}\wedge\delta A+\frac{\partial \textbf{L}}{\partial dA}\wedge \delta dA,$$
and then one integrates by parts to remove variations of the derivatives. This expression seems general and natural so I would expect it to hold also for non-volume forms. However, as one can see in the previous example, I am unable to do so.
Question
Is there an analogue result of the first variational formula for non-volume forms in field theory? Probably the mathematics of jet bundles are the correct setup for this question.
If the answer is positive, how can I write the variation of a Hodge dual as something proportional to $\delta A$.
NOTE:
I know that, given that field variations do not act on the coordinate functions, the variation of the Hodge dual can be written as $$\delta \star A=\star dx^\mu \delta A_\mu,$$ but I am interested in a formulation where $A$ is treated always as a differential form, and not using its components.