If I observe the distribution of the second order statistics $F_{X_{(2)}}(x)$, are there ways to back out the sample distribution $F_X(x)$, such that $F_x(x)$ satisfies the properties of a CDF.
I would say the inversion does not give a unique solution. Are there documented numerical ways (or approximated ones) to back out the family of $F_X(x)$?
One can write $F_{X_{(2)}}(x) = g(F_X(x))$ for a certain polynomial $g$, given in the first displayed equation in a Wikipedia article's section on the distribution functions of order statistics. It is not hard to see that the polynomial $g(t)$ is strictly increasing, $g(0)=0$ and $g(1)=1$. Hence solving $F_{X_{(2)}}(x) = g(F_X(x))$ for $F_X(x)$ given the value of $F_{X_{(2)}}(x)$ should always be possible, but (typically) not in "closed form".
(Note that $g(p)$ is the probability of the event that $Y\ge 2$, where $Y\sim\text{Bin}(n,p)$. Obviously $P(Y=0)=1$ when $p=0$ and $P(Y=n)=1$ when $p=1$, and so on. A simple coupling argument can show $g(t)\le g(u)$ if $t<u$.)