The existence of the inverse of a linear map is independent of the way the map affects the trace. Moreover, if an invertible map preserves a property then its inverse necessarily also preserves the property. Since depolarizing channel preserves the trace, so does its inverse.
Inverse of depolarizing channel
We can derive the formula for the inverse $\mathcal{D}_p^{-1}$ of a depolarizing channel $\mathcal{D}_p$ from the observation that the depolarizing parameter $p$ is multiplicative over channel composition, i.e. $\mathcal{D}_p\circ\mathcal{D}_q = \mathcal{D}_{pq}$. This implies that $\mathcal{D}^{-1}_p = \mathcal{D}_{1/p}$.
More explicitly, depolarizing channel $\mathcal{D}_p: L(\mathcal{H}) \to L(\mathcal{H})$ has the property that for any operator $X\in L(\mathcal{H})$
$$
\begin{align}
D_p(D_q(X)) &= p\left(qX + (1-q)\frac{\mathbb{I}}{d}\mathrm{tr}(X)\right) + (1-p)\frac{\mathbb{I}}{d}\mathrm{tr}(X) \\
&= pqX + (1-pq)\frac{\mathbb{I}}{d}\mathrm{tr}(X) \\
&= D_{pq}(X)
\end{align}
$$
where $d=\dim\mathcal{H}$. Therefore, if $p\ne 0$ then the inverse of $\mathcal{D}_p$ is $\mathcal{D}_{1/p}$ since then we have that
$$
\mathcal{D}_p \circ \mathcal{D}_{1/p} = \mathcal{D}_{1/p} \circ \mathcal{D}_p = \mathcal{D}_{p \cdot 1/p} = \mathcal{D}_1 = \mathcal{I}.
$$
If $p=0$, then $\mathcal{D}_0$ is constant on density operators and therefore does not have an inverse.
CP constraint
A linear map describes a physical process without post-selection, if it is completely positive (CP) and trace preserving (TP). Depolarizing channel $\mathcal{D}_p$ is completely positive if and only if $p\in\left[-\frac{1}{d^2-1}, 1\right]$. Moreover, if $p\in\left[-\frac{1}{d^2-1}, 1\right]$ and $\frac1p\in\left[-\frac{1}{d^2-1}, 1\right]$ then $p=1$. Therefore, the only completely positive depolarizing channel with completely positive inverse is the identity channel.
TP constraint
On the other hand, $\mathcal{D}_p$ preserves the trace for every $p$ and so the inverse $\mathcal{D}_{1/p}$ also preserves the trace. This is a special case of a general fact that if an invertible function $f$ preserves a property then its inverse $f^{-1}$ necessarily also preserves the property.