Notation: For a fixed function $f$ defined on $\mathbb{R}^{n}$ and $y\in\mathbb{R}^{n}$, we write $\tilde{f}(x)=f(-x)$ and $(\tau^{y}f)(x)=f(x-y)$.
I am assuming that you are familiar with the following result.
Theorem.
For $\phi\in\mathcal{S}(\mathbb{R}^{n})$, the tempered distribution $u\ast\phi$ coincides with a $C^{\infty}(\mathbb{R}^{n})$ function $\langle{u,\tau^{x}\tilde{\phi}}\rangle$ satisfying the growth estimates
$$\left|\partial^{\alpha}\langle{u,\tau^{x}\tilde{\phi}}\rangle\right|\leq C_{\alpha}(1+\left|x\right|)^{N_{\alpha}},\qquad\forall x\in\mathbb{R}^{n}$$
for all multi-indices $\alpha$. Moreover, if $u$ has compact support, then $u\ast\phi$ coincides with a Schwartz function.
And I leave it to you to prove the following lemma.
Lemma. Let $\phi\in\mathcal{S}(\mathbb{R}^{n})$ be such that $\int\phi=1$, and set $\phi_{\epsilon}:=\epsilon^{-n}\phi(\epsilon^{-1}\cdot)$. Then
$$f\ast\phi_{\epsilon}\rightarrow f \text{ in }\mathcal{S}(\mathbb{R}^{n}), \qquad\forall f\in\mathcal{S}(\mathbb{R}^{n})$$
as $\epsilon\downarrow 0$.
By duality, we see that for any tempered distribution $u$,
$$\langle{u\ast\phi_{\epsilon},f}\rangle=\langle{u,f\ast\widetilde{\phi_{\epsilon}}}\rangle=\langle{u,f\ast(\tilde{\phi})_{\epsilon}}\rangle\rightarrow \langle{u,f}\rangle$$
as $\epsilon\downarrow 0$. Whence, $u\ast\phi_{\epsilon}\rightarrow u$ in $\mathcal{S}'(\mathbb{R}^{n})$ as $\epsilon\downarrow 0$.
Define linear functionals $w_{1}$ and $w_{2}$ on $\mathcal{S}(\mathbb{R}^{n})$ respectively by
$$\langle{w_{1},\psi}\rangle:=\langle{u,\tilde{v}\ast\psi}\rangle, \quad \langle{w_{2},\psi}\rangle:=\langle{v,\tilde{u}\ast\psi}\rangle \qquad\forall \psi\in\mathcal{S}(\mathbb{R}^{n})$$
If I read your post correctly, you know how to verify that $w_{1}$ and $w_{2}$ are tempered distributions, so I will omit that detail. If we can show that $w_{1}\ast\phi=w_{2}\ast\phi$ in $\mathcal{S}'(\mathbb{R}^{n})$, then by replacing $\phi$ with $\phi_{\epsilon}$ and letting $\epsilon\downarrow 0$, we conclude that $w_{1}=w_{2}$ in the sense of tempered distributions.
Fix $\phi\in\mathcal{S}(\mathbb{R}^{n})$ and observe that for any $\psi\in\mathcal{S}(\mathbb{R}^{n})$,
\begin{align*}
\langle{w_{1}\ast\phi,\psi}\rangle=\langle{w_{1},\tilde{\phi}\ast\psi}\rangle&=\langle{u,\tilde{v}\ast(\tilde{\phi}\ast\psi)}\rangle\\
&=\langle{u,(\tilde{v}\ast\psi)\ast\tilde{\phi}}\rangle\\
&=\int_{\mathbb{R}^{n}}(u\ast\phi)(x)(\tilde{v}\ast\psi)(x)\mathrm{d}x\\
\end{align*}
where we use the commutativity of the convolution of ordinary functions, and the associativity of the convolution of a function and a distribution.
Similarly,
\begin{align*}
\langle{w_{2}\ast\phi,\psi}\rangle=\langle{v,\tilde{u}\ast(\tilde{\phi}\ast\psi)}\rangle=\langle{v,(\tilde{u}\ast\tilde{\phi})\ast\psi}\rangle&=\langle{v\ast\tilde{\psi},\tilde{u}\ast\tilde{\phi}}\rangle\\
\end{align*}
But
\begin{align*}
\int_{\mathbb{R}^{n}}(u\ast\phi)(x)(\tilde{v}\ast\psi)(x)\mathrm{d}x=\int_{\mathbb{R}^{n}}\langle{u,\tau^{x}\tilde{\phi}}\rangle\langle{v,\widetilde{\tau^{x}\tilde{\psi}}}\rangle\mathrm{d}x&=\int_{\mathbb{R}^{n}}\langle{u,\tau^{x}\tilde{\phi}}\rangle\langle{v,\tau^{-x}\psi}\rangle\mathrm{d}x\\
&=\int_{\mathbb{R}^{n}}\langle{u,\tau^{-x}\tilde{\phi}}\rangle\langle{v,\tau^{x}\psi}\rangle\mathrm{d}x\\
&=\int_{\mathbb{R}^{n}}\langle{u,\widetilde{\tau^{x}\phi}}\rangle\langle{v,\tau^{x}\psi}\rangle\mathrm{d}x\\
&=\langle{v\ast\tilde{\phi},\tilde{u}\ast\tilde{\psi}}\rangle
\end{align*}
This completes the proof.
