  The complete characterisation of the charge transport in a mesoscopic device
is provided by the Full Counting Statistics (FCS) $P_t(m)$, describing the
amount of charge $Q = me$ transmitted during the time $t$. Although numerous
systems have been theoretically characterized by their FCS, the experimental
measurement of the distribution function $P_t(m)$ or its moments $\langle Q^n
\rangle$ are rare and often plagued by strong back-action. Here, we present a
strategy for the measurement of the FCS, more specifically its characteristic
function $\chi(\lambda)$ and moments $\langle Q^n \rangle$, by a qubit with a
set of different couplings $\lambda_j$, $j = 1,\dots,k,\dots k+p$, $k = \lceil
n/2 \rceil$, $p \geq 0$, to the mesoscopic conductor. The scheme involves
multiple readings of Ramsey sequences at the different coupling strengths
$\lambda_j$ and we find the optimal distribution for these couplings
$\lambda_j$ as well as the optimal distribution $N_j$ of $N = \sum N_j$
measurements among the different couplings $\lambda_j$. We determine the
precision scaling for the moments $\langle Q^n \rangle$ with the number $N$ of
invested resources and show that the standard quantum limit can be approached
when many additional couplings $p\gg 1$ are included in the measurement scheme.
