The function you propose is related to the L'evy concentration function, studied by Kolmogorov, Rogozin, Esseen and others. See the special volume [1] https://link.springer.com/chapter/10.1007/978-94-011-2260-3_70
The classic book [2] has a chapter devoted to concentration functions with many references and the paper [3] has a quite sharp estimate; connection to combinatorics are in [4]. Also related is the study of small-ball probabilities, see the survey [5].
Returning to the original question, Decompose $\mu=\mu_a+\mu_s$ where $\mu_a$ is absolutely continuous to $\nu$ with Radon-Nikodym derivative $f$, and $\mu_s$ is singular to $\nu$. Write $M_s$ for the total mass of $\mu_s$. If $M_s \ge x$ then $F(x)=0$. Otherwise consider the sets $A_c:=\{f>c\}$. If there is such a set with $\mu(A_c)=x-M_s$ then $F(x)=\nu(A_c)$. If there is no such $c$ find the infimmum $c_*$ of the constants $c$ such that $\mu(A_c)<x-M_s$. one needs to do some tie-breaking inside the level set where $f=c_*$ and take a subset there of suitable $\mu$ measure $x-M_s-\mu(A_{c_*})$. Thus $F(x)=\nu(A_{c_*})+(x-\nu(A_{c_*}))/c_*$, I can add details if needed.
[1] Kruglov, V. M. "Concentration Functions (No. 45)." Selected Works of AN Kolmogorov. Springer, Dordrecht, 1992. 571-574.
[2] Petrov, Valentin Vladimirovich. Sums of independent random variables. Vol. 82. Springer Science & Business Media, 2012.
[3] Kesten, Harry. "A sharper form of the Doeblin–Levy–Kolmogorov–Rogozin inequality for concentration functions." Mathematica scandinavica 25.1 (1970): 133-144.
[4] Halász, Gábor. "Estimates for the concentration function of combinatorial number theory and probability." Periodica Mathematica Hungarica 8.3-4 (1977): 197-211.
[5] Li, Wenbo V., and Q-M. Shao. "Gaussian processes: inequalities, small ball probabilities and applications." Handbook of Statistics 19 (2001): 533-597.