There may be some clever trick to do this by elementary arguments, but I don't see it at the moment. With CFSG, the following argument shows that it is likely to be rare to have a Grothendieck ring with multiplicity $1$, if it happens at all. Let $b$ the the largest character degree of the non-Abelian finite simple group $G$, and suppose that $G$ has $k$ conjugacy classes. It is conjectured that $|G| <b^{3}$, and it has been proved by Cossey,Halasi,Maroti and Nguyen (using CFSG) that $|G| <b^{4}$. It has been proved by Fulman and Guralnick (again using CFSG) that $k < |G|^{0.41}$.
Your last inequality ( together with Cauchy_Schwarz) gives $b^{2} < \sqrt{k}\sqrt{|G|}$, so using the Fulman-Guralnick result, $b < |G|^{0.3525}$. Certainly $kb^{2} > |G|$, so we must have $k > |G|^{0.295}$. This last inequality can be achieved ( eg in ${\rm SL}(2,2^{n})$), but it seems likely to be relatively rare ( though precise checking may be painful).
Later edit: In fact, it is interesting to note that when $G = {\rm SL}(2,2^{n}) (n \geq 2)$, if we let $\chi$ denote the Steinberg character (which has degree $2^{n}$, whereas the largest irreducible character degree is $2^{n}+1$), we always .have $\chi \chi = \sum_{ \mu \in {\rm Irr}(G)} \mu$. This is because the projective cover of the (characteristic $2$) trivial module has dimension $2^{2n}-2^{n}$, and occurs as a summand of ${\rm St} \otimes {\rm St}.$