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 $pgg 1$ are included in the measurement scheme.
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