The first fusion ring is excluded from complex categorification in [LPR,$\S$14.1] using triangular prism equations. Note that the proof involves only the multiplicity one part of the fusion data, which is shared with the second fusion ring; thus, the second fusion ring is excluded as well.
Previously, the second fusion ring (only) was excluded from unitary categorification in [LPW,$\S$8.4] using quantum Fourier analysis. Here are some explanations:
Let $\mathcal{F}$ be a commutative fusion ring. The fusion matrices $(M_i)$ of $\mathcal{F}$ are commuting and normal, hence simultaneously diagonalisable. There exists an invertible matrix $P$ such that $P^{-1}M_iP = \mathrm{diag}(\lambda_{i,j})$. Let us denote $\Lambda=(\lambda_{i,j})$ as the character table of $\mathcal{F}$. Note that if $G$ is a finite group and $\mathcal{F}$ is the Grothendieck ring of $\mathrm{Rep}(G)$, then $\Lambda$ corresponds to the usual character table of $G$.
If $\mathcal{F}$ is the Grothendieck ring of a unitary fusion category and we choose $\lambda_{i,1}$ to be $\Vert M_i \Vert$, then by Corollary 8.5 in [LPW], for every triple $(j,k,\ell)$, we have: $$\sum_i \frac{\lambda_{i,j}\lambda_{i,k}\lambda_{i,\ell}}{\lambda_{i,1}} \ge 0.$$ Note that for $\mathrm{Rep}(G)$, this identity admits a combinatorial/probabilistic interpretation (see here).
Now, consider the character table of the second fusion ring mentioned above, specifically its last column: $$ \frac{1^3}{1} + \frac{0^3}{5} + \frac{0^3}{5} + \frac{0^3}{5} + \frac{1^3}{6} + \frac{(-3)^3}{7} + \frac{2^3}{7} = -\frac{65}{42} < 0.$$ This inequality implies that the second fusion ring does not admit a unitary categorification.
References
[LPR] Liu, Zhengwei; Palcoux, Sebastien; Ren, Yunxiang. Triangular prism equations and categorification, arXiv:2203.06522, 33 pages
[LPW] Liu, Zhengwei; Palcoux, Sebastien; Wu, Jinsong. Fusion bialgebras and Fourier analysis: analytic obstructions for unitary categorification. Adv. Math. 390 (2021), Paper No. 107905, 63 pp.