After some (formal!) manipulations I stumbled upon the following expression:
$$ \hat{f}\left(\xi,\eta\right)=\iint_{\mathbb{R}^{2n}}e^{2\pi i\left\langle x,t-\xi\right\rangle }e^{2\pi i\left\langle \left(P-Q\right)\left(x-\eta\right),t\right\rangle }\hat{g}\left(t,\left(P-Q\right)\left(x-\eta\right)+\eta\right)dtdx. $$ Notation: $\hat{f}$ is the Fourier transform, $\left\langle a,b\right\rangle $ is the inner product on $a,b\in\mathbb{R}^{n}$ and $P,Q$ are $n\times n$ matrices - for the sake of simplicity, at the beginning they can be taken as diagonal ones, namely $P=pI$, $Q=qI$, $p,q\in\mathbb{R}$ ($p=q$ eventually).
I am pretty sure that this expression hides a Fourier multiplier: in fact, I do expect it to boil down to something like (i.e. up to transposition) $$ \hat{f}\left(\xi,\eta\right)=e^{2\pi i\left\langle \left(P-Q\right)x,\eta\right\rangle }\hat{g}\left(\xi,\eta\right), $$ since for the diagonal case the formula should be precisely $$ \hat{f}\left(\xi,\eta\right)=e^{2\pi i\left(p-q\right)\left\langle x,\eta\right\rangle }\hat{g}\left(\xi,\eta\right), $$ but I am not able to reach that point. The integral representation of Dirac's delta seems involved (see the first exponential), but I cannot manage to exploit this connection. The expected formula and the initial expression agree for $P=Q$, which is a promising clue. From this case one sees that the role of "extra" exponential (i.e. the first in the integral expression) is related to the Fourier transform, but the main problem seems to be the second argument of $g$: it can be seen as translation, but I can't see how to come up with a useful insight.
Update
According to reuns' suggestion, I wrote $$ \hat{f}\left(\xi,\eta\right)=\int_{\mathbb{R}^{n\times n}}e^{-2\pi i\left\langle x,\xi\right\rangle }e^{2\pi i\left\langle t,x\right\rangle }G_{\eta}\left(x,t\right)dxdt,$$ where $$G_{\eta}\left(x,t\right)=e^{-2\pi i \left\langle \left(Q-P\right) x,t\right\rangle - \left\langle \left(Q-P\right) \eta,t\right\rangle }\hat{\rho}\left(t,\left(P-Q\right)x-\left(P-Q\right)\eta+\eta\right),$$ hence $$\hat{f}\left(\xi,\eta\right)=\left[\mathcal{F}_{x\rightarrow\xi}\left[\mathcal{F}_{t\rightarrow x}^{-1}G_{\eta}\left(x,t\right)\right]\left(x\right)\right]\left(\xi\right).$$ It is true that this coincides with $G_{\eta}(\xi,\xi)$? If yes, how does $$G_{\eta}\left(\xi,\xi\right)=e^{-2\pi i\left(\left\langle \left(Q-P\right)\xi,\xi\right\rangle -\left\langle \left(Q-P\right)\eta,\xi\right\rangle \right)}\hat{\rho}\left(\xi,\left(P-Q\right)\xi-\left(P-Q\right)\eta+\eta\right)$$ relate to the expected result?