= Solution
The $2\pi$ <periodization of a Schwartz function> is
$$
\boxed{P\phi(x)=\sum_{k\in\mathbb Z}\phi(x+2\pi k).}
$$
For every nonnegative integer $j$, the <Schwartz seminorms> give $|\phi^{(j)}(x+2\pi k)|\leq C_j(1+|k|)^{-2}$ uniformly for $x\in[0,2\pi]$. The <Weierstrass M-test> gives absolute and <uniform convergence> there of the defining <series> and all its differentiated <series>. Termwise differentiation proves that $P\phi$ is a <smooth function>, with $(P\phi)^{(j)}=P(\phi^{(j)})$. Reindexing the absolutely convergent <series> proves $P\phi(x+2\pi)=P\phi(x)$, so it is a <periodic function>.
For an explicit <smooth periodic partition of unity>, choose the <bump function>
$$
\eta(x)=\begin{cases}
\exp\!\bigl(-1/(1-(x/(2\pi))^2)\bigr),&|x|<2\pi,\\
0,&|x|\geq2\pi.
\end{cases}
$$
It is a <test function>, positive on $[-\pi,\pi]$. The <smooth function> $S(x)=P\eta(x)$ is $2\pi$ periodic and strictly positive, since every $x$ has a translate in $[-\pi,\pi]$. Set
$$
\boxed{\Psi(x)=\frac{\eta(x)}{S(x)}\in C_c^\infty(\mathbb R),\qquad
P\Psi(x)=\frac{\sum_k\eta(x+2\pi k)}{S(x)}=1.}
$$
Division preserves smoothness because $S>0$, and the <support> remains contained in $[-2\pi,2\pi]$.
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