= Solution
For $\delta>0$ and $M=\|f''\|_\infty>0$, differentiate the bound $F(h)=\sqrt6\delta/h+Mh/2$. Its derivative vanishes at $h_*^2=2\sqrt6\delta/M$, and $F''(h)>0$. Consequently, when $h_*<1/2$,
$$
\boxed{h_* =\left(\frac{2\sqrt6\delta}{M}\right)^{1/2},\qquad F(h_*)=\sqrt{2\sqrt6 M\delta}.}
$$
If $h_*\geq1/2$, the infimum on the open allowed interval is approached as $h\uparrow1/2$, rather than attained at an admissible endpoint. If $M=0$ and $\delta>0$, the bound decreases throughout the interval and has the same boundary behavior. At zero noise with $M>0$, its infimum is approached as $h\downarrow0$.
A practical <regularization parameter choice> independent of $M$ is
$$
\boxed{h(\delta)=\min\{1/4,\delta^\theta\},\qquad0<\theta<1.}
$$
It satisfies $h\to0$ and $\delta/h\to0$, hence gives a <convergent regularization of an inverse problem> on the domain of the <Volterra operator> inverse. This extends beyond $C^2$ exact data: if $f=Ku$ with $u\in L^2$, the differences are local averages of $u$. Extend $u$ by zero outside the interval; continuity of translations in $L^2$ makes these averages converge to $u$ on both halves. Together with $\|D_h(f^\delta-f)\|_2\leq\sqrt6\delta/h$ this proves convergence uniformly over the noise ball for every admissible exact datum.
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