= Solution
Use the <Gaussian curvature> $-1$ normalization
$$
ds^2=\frac{dx^2+dy^2}{y^2}
$$
of the <Poincaré half-plane model>. This is a complete <conformal metric>, invariant under real <Möbius transformations>. Its quotient by the free effective <principal congruence subgroup> is again complete: a <geodesic> lifts to the complete covering plane and extends there for all time. Transporting the quotient metric through the <biholomorphism> supplies the required complete <conformal metric>.
A nonconstant closed <geodesic> corresponds to a <hyperbolic Möbius transformation> $A$ of $\Gamma(2)$, and its <hyperbolic translation length> is $2\operatorname{arcosh}(|\operatorname{tr}A|/2)$. Indeed its <eigenvalues> have magnitudes $\lambda,\lambda^{-1}$ with $\lambda>1$, and its axis quotient has length $2\log\lambda$. For $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ in $\Gamma(2)$, $bc\equiv0\pmod4$ implies $ad\equiv1\pmod4$. The odd numbers $a,d$ therefore have the same residue modulo four, and
$$
\operatorname{tr}A=a+d\equiv2\pmod4.
$$
A <hyperbolic Möbius transformation> must have $|\operatorname{tr}A|>2$, so its smallest possible absolute <trace> is six. It is attained by
$$
\begin{pmatrix}1&2\\0&1\end{pmatrix}
\begin{pmatrix}1&0\\2&1\end{pmatrix}
=\begin{pmatrix}5&2\\2&1\end{pmatrix}.
$$
Consequently
$$
\boxed{\ell_{\min}=2\operatorname{arcosh}3=2\log(3+2\sqrt2).}
$$
This element is primitive: a proper power would have a shorter root represented by a <hyperbolic Möbius transformation>, contradicting the <trace> bound. No essential simple closed <geodesic> exists on the three-punctured sphere, since every essential simple loop is peripheral and corresponds to a <parabolic Möbius transformation>. The minimizing closed <geodesic> is therefore nonsimple. Finally, the PDF leaves the numerical <Gaussian curvature> unspecified: the displayed length uses <Gaussian curvature> $-1$; for <Gaussian curvature> $-\kappa^2$, it is divided by $\kappa$.
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