Obviously as a starting point to this kind of question, we must recognise that there are may possible probability distributions on the relevant support that have the same (finite) set of moments. Here you have, say, 101 possible outcomes for the grade but only two fixed moments, so you have two equations in one-hundred unknowns (once we subtract a degree-of-freedom for the norming constraint of probability). If you add in the additional constraints from the order statistics then you have five equations in one-hundred unknowns This means that any method to construct a "simplest" distribution is typically going to involve some method of quantifying the "simplicity" of any possible distribution and then an optimisation of that quantity over the possible distributions.
One well-known approach to this problem is the technique of finding the maximum entropy distribution given a set of moment constraints (and the support for the random variable). In this method, the distribution is chosen by maximising the entropy under the given constraints. Since entropy represents the amount of "information" or "uncertainty" in a distribution, you might reasonably regard this as a proxy for "simplicity".
Suppose that the possible marks on this test are $0,...,M$ and you have $n$ students, so that the sample distribution can be represented from a count vector $\mathbf{n} = (n_0,...,n_M)$ with $\sum n_i = n$. The entropy of the distribution is:
$$H(\mathbf{p}) \equiv \sum_{i=0}^M \frac{n_i}{n} \cdot \log \bigg( \frac{n_i}{n} \bigg).$$
Your sample constraints are knowledge of the statistics $x_{(1)}$, $x_{((n+1)/2)}$, $x_{(n)}$, $\bar{x}_n$ and $s_n^2$. All of these can be written as functions of the count vector $\mathbf{n}$, which means that you have five constraining equations on the count vector.$\dagger$ To find the maximum entropy distribution, your optimisation problem would be:
$$\text{Maximise } H(\mathbf{p}) \text{ subject to constraints }$$
Unfortunately, since four of your five constraints are nonlinear, this is an extremely complicated nonlinear discrete optimisation problem. Unless the sample size is small, it may be extremely difficult to compute any count vector that exactly satisfies the contraints, let alone find if there is more than one point in the optimisation set. Consequently, you will need to give consideration to finding an algorithm to maximise the entropy subject to those constraints and you will need to decide whether you want to relax things (perhaps by looking at a continuous approximation that allows noninteger counts) to simplify the optimisation problem. In any case, setting aside the computational difficulties of the optimisation problem, maximum-entropy is one reasonable way to optimise for a "representative" distribution that obeys the relevant constraints.
$\dagger$ Here I have assumed an odd-number of data points for simplicity in computing the median. You will need to make appropriate changes to the median formula if there are an even number of data points.
