Changing variables, we have
$$(pv(K)\ast f)(x)=\lim_{\delta\searrow 0}
\int_{\vert x-y\vert>\delta}K(x-y) f(y) dy=\lim_{\delta\searrow 0}
\int_{\vert x-G(w)\vert>\delta}K(x-G(w)) f(G(w))\vert \nabla G(w)\vert dw,
$$
so that
$$
(pv(K)\ast f)(G(\nu))=\lim_{\delta\searrow 0}\int_{\vert G(\nu)-G(w)\vert>\delta}K(G(\nu)-G(w)) f(G(w))\vert \nabla G(w)\vert dw.
$$
Now, if $G$ is globally Lipschitz continuous, i.e. $\nabla G\in L^\infty$, we have
$$
\{w,\vert G(\nu)-G(w)\vert>\delta\}\subset\{w,\Vert\nabla G\Vert_{L\infty}\vert\nu-w\vert>\delta\}.
$$
Since $\nabla G\circ \nabla G^{-1}=Id$, we have
$
\Vert\nabla G\Vert\Vert \nabla G^{-1}\Vert\ge 1.
$
As a result, if $G$ is globally bi-Lipschitz continuous (i.e. $G,G^{-1}$ are both globally Lipschitz continuous), we find $\alpha\ge 1$ such that
$$
\{w,\vert w-v\vert\le \delta/\ \alpha\}\subset \{w,\vert G(w)-G(v)\vert\le \delta\}\subset
\{w,\vert w-v\vert\le \alpha\delta\}.\tag E
$$
We may now consider what is now a general singular integral, and not only a Fourier multiplier,
that is the operator with kernel
$$
k(v,w)=K(G(v)-G(w))\vert\nabla G(w)\vert.
$$
We have indeed the following estimates
$$
\vert k(v,w)\vert\lesssim\vert v-w\vert^{-n},\quad\vert\partial_vk(v,w)\vert+\vert\partial_wk(v,w)\vert\lesssim\vert v-w\vert^{-n-1}
\tag {CZ}$$
$$
(pv(K)\ast f)(G(\nu))=\lim_{\delta\searrow 0}\int_{\vert G(\nu)-G(w)\vert>\delta}K(G(\nu)-G(w)) f(G(w))\vert \nabla G(w)\vert dw.
$$
Using the embeddings (E), we see that that $\{w,\vert G(w)-G(v)\vert> \delta\}$ is the union of a set $\{w,\vert w-v\vert> \alpha\delta\}$ with a set where $\vert G(w)-G(v)\vert\sim \delta\sim \vert v-w\vert$ with volume $\delta^{n}$ on which $K$ is of size $\delta^{-n}$.
From the Lebesgue differentiation theorem, this part of the integral converges as well.
$$
(pv(K)\ast f)(G(\nu))=\lim_{\delta\searrow 0}\int_{\vert\nu-w\vert>\delta}K(G(\nu)-G(w)) f(G(w))\vert \nabla G(w)\vert dw+\Omega f,
$$
where $\Omega$ is $L^p$ bounded for $p\in(1,+\infty)$.
So when you change variables in a singular integral appearing as a Fourier multiplier or a convolution, you get a more general type of operator, a Calder\'on-Zygmund type of operator
with a kernel satisfying (CZ).