I wouldn't say it counts as well known, but if you have a look at L. Schwartz' Théorie des distributions, chap. VI, § 7 and § 8, you can observe an interesting trick involving fundamental solutions of differential operators that is used several times to deduce properties of distributions from the properties of their regularisations.
A general result
Theorem. — Let $K$ be a compact neighbourhood of $0$ in $\mathbb R^n$ and let $F$ be a subspace of $\mathcal D'(\mathbb R^n)$ such that for every $T\in F$ and every $f\in\mathcal D(\mathbb R^n)$ supported in $K$, the smooth function $T*f$ belongs to $F$. Suppose that there is a norm $\|\cdot\|$ on $F$ for which it is a Banach space with a topology finer than the one induced by $\mathcal D'(\mathbb R^n)$. Let $T\in\mathcal D'(\mathbb R^n)$. The following conditions are equivalent:
- $T$ is a linear combination of derivatives of elements of $F$.
- For every $f\in\mathcal D(\mathbb R^n)$ supported in $K$, the smooth function $T*f$ belongs to $F$.
The proof below is a mix of the proofs of chap. VI, § 7, Theorem XXII and § 8, Theorem XXV in L. Schwartz' book, where the trick is to bring into the picture convolutions with functions of class $\mathrm C^m$ for some $m\in\mathbb N$ rather than smooth functions and use a fundamental solution of class $\mathrm C^m$ of some differential operator to convolve with $T$.
Proof of the theorem. — (1$\Rightarrow$2) Suppose that there are finite families $(T_i)_{i\in I}$ and $(\alpha_i)_{i\in I}$ of elements of $F$ and $\mathbb N^n$ respectively such that $T=\sum_{i\in I}\partial^{\alpha_i}T_i$. If $f\in\mathcal D(\mathbb R^n)$ is supported in $K$, then $$T*f=\sum_{i\in I}\partial^{\alpha_i}T_i*f=\sum_{i\in I}T_i*\partial^{\alpha_i}f$$ which shows that $T*f\in F$ since each $\partial^{\alpha_i}f$ is smooth and supported in $K$.
(2$\Rightarrow$1) Suppose that $T*f\in F$ for every $f\in\mathcal D(\mathbb R^n)$ supported in $K$.
Step 1. There exists an open neighbourhood of $0$ in $K$ as well as a $m\in\mathbb N$ such that for every function $f$ on $\mathbb R^n$ of class $\mathrm C^m$ supported in $U$, the distribution $T*f$ belongs to $F$.
Denote by $\mathcal D_K(\mathbb R^n)$ be the space of all smooth functions on $\mathbb R^n$ supported in $K$ endowed with the topology of "uniform convergence of all derivatives", making it a Fréchet space. The closed graph theorem tells you that the linear map $f\mapsto T*f$ from $\mathcal D_K(\mathbb R^n)$ to $F$ is continuous. There is therefore a $m\in\mathbb N$ as well as a $r>0$ such that for every $f\in\mathcal D_K(\mathbb R^n)$, $$ \sup_{|\alpha|\leq m}\sup_{x\in\mathbb R^n}|\partial^\alpha f(x)|\leq r\implies\|T*f\|\leq 1. $$ Let $U$ be an open neighbourhood of $0$ in $\mathbb R^n$ contained in $K$ and denote by $\mathcal D_U^m(\mathbb R^n)$ the space of function on $\mathbb R^n$ of class $\mathrm C^m$ and supported in $U$ endowed with the norm $$ f\mapsto\sup_{|\alpha|\leq m}\sup_{x\in\mathbb R^n}|\partial^\alpha f(x)|. $$ It is clear that $\mathcal D_K(\mathbb R^n)\cap\mathcal D_U^m(\mathbb R^n)$ is dense in $\mathcal D_U^m(\mathbb R^n$, hence the existence of a continuous linear map $u$ from $\mathcal D_U^m(\mathbb R^n)$ to $F$ such that $u(f)=T*f$ for every $f\in\mathcal D_K(\mathbb R^n)\cap\mathcal D_U^m(\mathbb R^n)$. The linear maps $f\mapsto u(f)$ and $f\mapsto T*f$ from $\mathcal D_U^m(\mathbb R^n)$ to $\mathcal D'(\mathbb R^n)$ are then continuous, coincide on a dense set and therefore must be equal.
Step 2. There are functions $f$ and $g$ on $\mathbb R^n$ of class $\mathrm C^m$ supported in $U$ as well as a translation invariant differential $\Psi$ such that $f+\Psi(g)=\delta_0$.
Let $E$ be the function $(t_1,t_2,\dots,t_n)\mapsto (m+1)!^{-n}(t_1t_2\cdots t_n)^{m+1}H(t_1)H(t_2)\cdots H(t_n)$ (where $H$ is the Heaviside function) and let $\Psi$ be the differential operator $(\partial_1\partial_2\cdots\partial_n)^{m+2}$. It easy to check that $\Psi(E)=\delta_0$ and that $E$ is of class $\mathrm C^m$ on $\mathbb R^n$ and of class $\mathrm C^\infty$ on $\mathbb R^n\setminus\{0\}$. Let $\varphi\in\mathcal D_U^m(\mathbb R^n)$ be such that $\varphi=1$ on a neighbourhood of $0$ in $U$. The function $g=\varphi E$ obviously belongs to $\mathcal D_U^m(\mathbb R^n)$. The function $f=\Psi((1-\varphi)E)$ is then obviously smooth and the relation $$ \Psi((1-\varphi)E)+\Psi(\varphi E)=\Psi(E)=\delta_0 $$ show that $f$ is supported in $U$ (since $\Psi(\varphi E)$ and $\delta_0$ are) and that $f+\Psi(g)=\delta_0$.
Step 3. Conclusion.
Notice that $ T=T*\delta_0=T*f+T*\Psi(g)=T*f+\Psi(T*g)$. Since $f,g\in\mathcal D_U^m(\mathbb R^n)$, you have $T*f\in F$ and $T*g\in F$ and this shows that $T$ is indeed a linear combination of derivatives of elements of $F$.
Relation to your question
Thanks to the general result above, you can see your question boils down to finding a good Banach space of compactly supported distributions.
Lemma. — Let $K$ be a compact neighbourhood of $0$ in $\mathbb R^n$ and let $T\in\mathcal D'(\mathbb R^n)$ be such that for every $f\in\mathcal D(\mathbb R^n)$ supported in $K$, the smooth function $T*f$ is compactly supported. There exists a compact subset $L$ of $\mathbb R^n$ such that for every $f\in\mathcal D(\mathbb R^n)$ supported in $K$, the smooth function $T*f$ is supported in $L$.
Hence, you can for example take $F$ to be the Banach space of all continuous functions supported in $L$ with the norm of uniform convergence, and the previous theorem tells you that $T$ is a linear combination of derivatives of compactly supported continuous functions and is therefore compactly supported.
Proof of the lemma. — Once again, denote by $\mathcal D_K(\mathbb R^n)$ be the space of all smooth functions on $\mathbb R^n$ supported in $K$ endowed with the topology of "uniform convergence of all derivatives", for which it is a Fréchet space. For every $k\in\mathbb N$, let $F_k$ be the closed subspace of $\mathcal D(\mathbb R^n)$ consisting of the smooth functions on $\mathbb R^n$ supported in $[-k,k]$ and let $E_k$ be the subspace of $\mathcal D_K(\mathbb R^n)$ preimage of $F_k$ by $f\mapsto T*f$. There are some general forms of the closed graph theorem that tell you that the map $f\mapsto T*f$ from $\mathcal D_K(\mathbb R^n)$ to $\mathcal D(\mathbb R^n)$ is continuous. Hence, each $E_k$ is closed in $\mathcal D_K(\mathbb R^n)$. Moreover, since the subspaces $F_k$ cover $\mathcal D(\mathbb R^n)$, the subspaces $E_k$ cover $\mathcal D_K(\mathbb R^n)$ and since $\mathcal D_K(\mathbb R^n)$ is a Baire space, at least one of the $E_k$ has non-empty interior in $\mathcal D_K(\mathbb R^n)$ and therefore is equal to $\mathcal D_K(\mathbb R^n)$.
