All the basic notions for a fusion ring are defined in this post.
The fusion ring $G_p$ is the Grothendieck ring of the cyclic group of prime order $p$.

Let $\mathcal{F}$ be a fusion ring with distinguished basis $\{h_1, h_2, \dots, h_r \}$ and fusion rules $$ h_i \cdot h_j = \sum_k n_{ij}^kh_k. $$ Definition: The fusion ring $\mathcal{F}$ is multiplicity one if every $ n_{ij}^k \in \{0,1\}$.

Lemma: Let $\mathcal{F}$ be a fusion ring of multiplicity one and rank $r$, then FPdim$(\mathcal{F}) \le r^3$.
Proof: $d(h_i)^2 = \sum_k n_{ii}^kd(h_k) \le \sum_k d(h_k) \le \sum_k (\sum_s d(h_s))^{1/2} = r (\sum_s d(h_s))^{1/2}$
Let $x = \sum_k d(h_k)>0$. Then, $x^2 \le r^2x$, and so $x \le r^2$. It follows that $d(h_i) \le r$.
But FPdim$(\mathcal{F}) = \sum_i d(h_i)^2 \le \sum_i r^2 = r^3$. $\square$

Digression: at multiplicity $m$, we get idem that $d(h_i) \le mr$ and FPdim$(\mathcal{F}) \le m^2r^3$.

Theorem: There is no integral simple fusion ring of multiplicity one and rank $\le 10$ (except $G_p$).
Proof: By the previous proposition, a fusion ring of multiplicity one and rank $\le 10$, has FPdim $ \le 10^3$.
But by a SAGE computation (with this code), there is no integral simple fusion ring of multiplicity one, rank $\le 10$ and FPdim $ \le 1000$ (except $G_p$). $\square$

Question: Is there an integral simple fusion ring of multiplicity one (except $G_p$)?