Suppose $f$ is a normalized cuspidal eigenform of level $p^2N$ ($p\nmid N$) and trivial character, such that the corresponding representation at $p$ is supercuspidal. Now suppose $\chi$ is primitive Hecke character of conductor $p$. We can apply the usual twisting operator by $\chi$ or $\chi^{-1}$ to $f$ to obtain normalized eigenforms $f_\chi$ and $f_{\chi^{-1}}$ with character $\chi^2$ and $\chi^{-2}$ respectively. Now we consider the adelic Atkin-Lahner operator given by the matrix $\begin{pmatrix} 0 & 1 \\ p^2 & 0\end{pmatrix}$ at $p$. Then one sees that it maps $f_\chi$ to $f_{\chi^{-1}}\otimes(\chi^2\circ\det)$ with some scalar multiple. My question is what is this scalar multiple? In particular is it a $p$-adic unit? (edited based on comment)

What you are asking for is a formula for the local epsilon-factors $\varepsilon(\pi \otimes \chi)$ where $\pi = \pi_{f, p}$ is the local component of $f$ at $p$. This is a deep question: it has to be, in some sense, since you can recover $\pi$ uniquely if you know the epsilon-factors of all its twists (Jacquet's local converse theorem).

Anyway, since you are assuming that $f$ has level $Np^2$ and trivial character, the representation $\pi$ is not too nasty: it's a "depth 0 supercuspidal", arising from a character $\eta: \mathbf{F}_{p^2}^\times \to \mathbf{C}^\times$ trivial on $\mathbf{F}_p^\times$. There is a paper by Jared Weinstein and me which gives an algorithm for computing $\eta$. Once you have this, there is a formula for the epsilon-factors in terms of Gauss sums over $\mathbf{F}_{p^2}^\times$. Up to some normalisation factor, $\varepsilon(\pi \otimes \chi)$ will be something like $\sum_{x \in \mathbf{F}_{p^2}^\times} \eta(x) \chi(\mathrm{Nm}(x)) e^{2\pi i \mathrm{Tr}(x)/p}$.