In preparing the material for asking the question, I understood it better
and ended out figuring out an answer it.
As answering your own question
is encouraged in MathOverflow, I'm posting it here. The proof relies on the fact
that, using Levy's modulus of continuity, when restricting $W_t$
to pass between a series of hoops at discrete time points it is contained within
a larger tube. As both the hoop diameter and density of the mesh vanish,
$\sup_{t\in[0,1]}|W_t|\to 0$.
Under this double limit the standard derivation of the Onsager--Machlup
functional still holds.
Together with the fact that, for each fixed partition, the normalized hoop
probability converges to the density, we have that the double limit
is equal to
the iterated limit, which is the limit of densities as the partition
mesh vanishes.
$
\newcommand{\limsup}{\operatorname*{lim\,sup}}
\newcommand{\reals}{{\rm I\!R}}
\newcommand{\naturals}{\mathbb{N}}
\newcommand{\parti}{\mathcal{P}}
\newcommand{\xeballi}{%
\max_{\tau\in\mathcal{P}_i} |X_\tau-\phi(\tau)|<\epsilon %
}
\newcommand{\weballi}{%
\max_{\tau\in\mathcal{P}_i} |W_\tau|<\epsilon %
}
\newcommand{\xeiball}{%
\max_{\tau\in\mathcal{P}_i} |X_\tau-\phi(\tau)|<\epsilon_i %
}
\newcommand{\weiball}{%
\max_{\tau\in\mathcal{P}_i} |W_\tau|<\epsilon_i %
}
$
In what follows, let $\{\parti_i\}_{i=1}^\infty$ be a sequence of
partitions $\parti_i:=\{\tau_{k,i}\}_{k=0}^{N_i}$ of the unit interval
whose mesh $\bar\delta_i$ vanishes, i.e.,
$$
\begin{align}
\tau_{0,i}&=0,&
\tau_{N_i,i}&= 1,&
\tau_{k,i}&<\tau_{k+1,i},\\
\end{align}
$$
$$
\begin{align}
\delta_{k,i} &:= \tau_{k,i} - \tau_{k-1,i},&
\bar\delta_i &:= \max_{0< k\leq N_i} \delta_{k,i}, &
\lim_{i\to\infty}\bar\delta_i &= 0.
\end{align}
$$
Lemma 1: For all $i \in\naturals$ and $\phi:[0,1]\to\reals^n$,
$$
\lim_{\epsilon\to 0}
\frac{
P(\max_{\tau\in\mathcal{P}_i} |X_\tau-\phi(\tau)|<\epsilon)
}{
\mu(\{x\in\reals^n:|x|<\epsilon\})^{N_i}
} = \prod_{k=1}^{N_i}
p\big(\phi(\tau_{k-1,i});\delta_{ki}, \phi(\tau_{ki})\big),
$$
where $\mu$ is the Lebesgue measure.
This is a corollary of the Lebesgue differentiation theorem and the fact that
$X_\tau$ admits a continuous probability density function.
Lemma 2: For all $\phi:[0,1]\to\reals^n$ in the Cameron--Martin space,
$$
\lim_{\substack{\epsilon\to 0 \\ i\to\infty}}
\frac{
P(\max_{\tau\in\mathcal{P}_i} |X_\tau-\phi(\tau)|<\epsilon)
}{
P(\max_{\tau\in\mathcal{P}_i} |W_\tau|<\epsilon)
} =
\exp\big(J(\phi)\big).
$$
This is follows from the same arguments that are used to prove the
Onsager--Machlup functional as the fictitious density of tubes around $\phi$,
which will be detailed shortly.
Corollary 3: For all $\phi:[0,1]\to\reals^n$ in the Cameron--Martin space,
$$
\lim_{i\to\infty}
\prod_{k=1}^{N_i}
\frac{
p\big(\phi(\tau_{k-1,i});\delta_{ki}, \phi(\tau_{ki})\big)
}{(2\pi)^{-\frac n2}(\delta_{ki})^{-\frac12}} =
\exp\big(J(\phi)\big).
$$
Proof:
Applying Lemma 1 to both the $X_t$ and $W_t$ processes, we have that, for all
$i\in naturals$,
$$
\lim_{\epsilon\to 0}
\frac{
P(\max_{\tau\in\mathcal{P}_i} |X_\tau-\phi(\tau)|<\epsilon)
}{
P(\max_{\tau\in\mathcal{P}_i} |W_\tau|<\epsilon)
} = \prod_{k=1}^{N_i}
\frac{
p\big(\phi(\tau_{k-1,i});\delta_{ki}, \phi(\tau_{ki})\big)
}{(2\pi)^{-\frac n2}(\delta_{ki})^{-\frac12}}.
$$
As the double limit of Lemma 2 exists, and the single limit for each
fixed $i\in\naturals$ exists and is finite,
we have that
$$
\begin{equation}
\tag*{$\blacksquare$}
\lim_{\substack{\epsilon\to 0 \\ i\to\infty}}
\frac{
P(\max_{\tau\in\mathcal{P}_i} |X_\tau-\phi(\tau)|<\epsilon)
}{
P(\max_{\tau\in\mathcal{P}_i} |W_\tau|<\epsilon)
} =
\lim_{i\to\infty}
\lim_{\epsilon\to 0}
\frac{
P(\max_{\tau\in\mathcal{P}_i} |X_\tau-\phi(\tau)|<\epsilon)
}{
P(\max_{\tau\in\mathcal{P}_i} |W_\tau|<\epsilon)
}.
\end{equation}
$$
Now only Lemma 2 remains to be proved. For that, we will the following lemma,
which is a consequence of Levy's modulus of continuity theorem.
Lemma 4:
There exist $c\in\reals$ such that, for all sufficiently large $i$ and almost
all outcomes in the event $\{\weballi\}$,
$$
|W_t| < \epsilon + c\sqrt{\bar \delta_i\log(1/\delta_i)},
\qquad\forall t\in[0,1].
$$
The proof of Lemma 2 now follows essentially the same steps as the derivation of
the Onsager--Machlup functional (see, e.g., Capitaine, 2000).
Proof of Lemma 2:
As we are dealing with a double limit, let $\{\epsilon_i\}_{i=1}^\infty$ be a
sequence of vanishing hoop diameters $\epsilon_i\to 0$, indexed by the same $i$
as the partitions. Applying the Girsanov transformation,
$$
\begin{multline}
\frac{\xeiball}{\weiball} = \exp\big(J(\phi)\big)
E\Bigg[\exp\Big(
-\int_0^1\dot\phi_t^T\,dW_t
\\
-\frac12\int_0^1|f(\phi_t+W_t)|^2\,dt + \frac12\int_0^1|f(\phi_t)|^2\,dt
\\
+\int_0^1 f(\phi_t+W_t)^T d\phi_t - \int_0^1 f(\phi_t)^T d\phi_t
\\
+\int_0^1 f(\phi_t+W_t)^T d\phi_t +
+\frac12 \int_0^1\operatorname{div}f(\phi_t)\,dt
\Big)\Bigg|\weiball\Bigg]
\end{multline}
$$
It now suffices to prove that, for all $c\in\reals$,
$$
\begin{equation}
\tag{1}
\limsup_{i\to\infty}
E\Bigg[\exp\Big(c
\int_0^1\dot\phi_t^T\,dW_t
\Big)\Bigg|\weiball\Bigg]\leq 1,
\end{equation}
$$
$$
\begin{equation}
\tag{2}
\limsup_{i\to\infty}
E\Bigg[\exp\Big(
c\frac12\int_0^1|f(\phi_t+W_t)|^2\,dt - c\frac12\int_0^1|f(\phi_t)|^2\,dt
\Big)\Bigg|\weiball\Bigg]\leq 1,
\end{equation}
$$
$$
\begin{equation}
\tag{3}
\limsup_{i\to\infty}
E\Bigg[\exp\Big(
c\int_0^1 f(\phi_t+W_t)^T d\phi_t - c\int_0^1 f(\phi_t)^T d\phi_t
\Big)\Bigg|\weiball\Bigg]\leq 1,
\end{equation}
$$
$$
\begin{equation}
\tag{4}
\limsup_{i\to\infty}
E\Bigg[\exp\Big(
c\int_0^1 f(\phi_t+W_t)^T dW_t
+c\frac12 \int_0^1\operatorname{div}f(\phi_t)\,dt
\Big)\Bigg|\weiball\Bigg]\leq 1,
\end{equation}
$$
Eqs. (2) and (3) are easy to verify, as $f$ is assumed to be Lipschitz and
bounded.
Eq. (4) follows from the application of the stochastic Stokes' theorem.
Eq. (1) is proved by Shepp and Zeitouni (1992) for conditioning on
the events $\{\sup_{t\in[0,1]}|W_t|<\epsilon\}$.
However, their proof holds for the events $\{\weiball\}$, as they are symmetric
and convex, and Lemma 4 implies that the width of the bounding tubes vanish
as $i\to\infty$.
