Joel Kamnitzer explained that $\omega_k$ is such a fundamental representation iff $0$ is a weight of the corresponding representation. In this answer I want to explain that such a weight exists for an irreducible root system exactly when the root system in not of Type A.

Note that the highest short root $\widehat{\theta}$ is the minimal (in root order) dominant weight greater than $0$ in root order. In particular, $0$ is a weight of $V(\widehat{\theta})$. (This representation is called a "quasi-minuscule representation"; see https://en.wikipedia.org/wiki/Minuscule_representation.) Thus, if the highest short root happens to coincide with a fundamental weight $\omega_k$, then it will satisfy your requirement that there is some $V$ for which $V$ belongs to $V\otimes V(\omega_k)$.

I claim that for every irreducible root system other than Type A, we indeed have that $\widehat{\theta}$ is a fundamental weight.

For simply laced root systems, $\widehat{\theta}$ is the same as the highest root $\theta$, and there is a nice combinatorial rule to write the coefficients of $\theta$ in the basis of fundamental weights: writing $\theta = \sum_{i=1}^{n}c_i\omega_i$, the coefficient $c_i$ is the number of edges between the node numbered $i$ and the special affine node in the extended Dynkin diagram. (This is in Bourbaki somewhere.) Then we can just check the extended Dynkin diagrams (https://en.wikipedia.org/wiki/Dynkin_diagram#Affine_Dynkin_diagrams) and see that the simply laced extended Dynkin diagrams have a single edge adjacent to their affine nodes, except in Type A.

I think there's an extension of this combinatorial rule to non-simply laced cases and for the highest short root using some variant of extended Dynkin diagrams, but I don't remember exactly. So let me just say that it's not hard to check, using say the standard realizations, that for $B_n$ the highest short root is $\omega_2$, and for $C_n$ the highest short root is $\omega_1$. And similarly it can be checked that for $G_2$ the highest short root is $\omega_1$, and for $F_4$ the highest short root is $\omega_1$.

Joel Kamnitzer explained that $\omega_k$ is such a fundamental representation iff $0$ is a weight of the corresponding representation. In this answer I want to explain that such a weight exists for an irreducible root system exactly when the root system in not of Type A.

Note that the highest short root $\widehat{\theta}$ is the minimal (in root order) dominant weight greater than $0$ in root order. In particular, $0$ is a weight of $V(\widehat{\theta})$. (This representation is called a "quasi-minuscule representation"; see https://en.wikipedia.org/wiki/Minuscule_representation.) Thus, if the highest short root happens to coincide with a fundamental weight $\omega_k$, then it will satisfy your requirement that there is some $V$ for which $V$ belongs to $V\otimes V(\omega_k)$.

I claim that for every irreducible root system other than Type A, we indeed have that $\widehat{\theta}$ is a fundamental weight.

For simply laced root systems, $\widehat{\theta}$ is the same as the highest root $\theta$, and there is a nice combinatorial rule to write the coefficients of $\theta$ in the basis of fundamental weights: writing $\theta = \sum_{i=1}^{n}c_i\omega_i$, the coefficient $c_i$ is the number of edges between the node numbered $i$ and the special affine node in the extended Dynkin diagram. (This is in Bourbaki somewhere.) Then we can just check the extended Dynkin diagrams (https://en.wikipedia.org/wiki/Dynkin_diagram#Affine_Dynkin_diagrams) and see that the simply laced extended Dynkin diagrams have a single edge adjacent to their affine nodes, except in Type A.

I think there's an extension of this combinatorial rule to non-simply laced cases and for the highest short root using some variant of extended Dynkin diagrams, but I don't remember exactly. So let me just say that it's not hard to check, using say the standard realizations, that for $B_n$ the highest short root is $\omega_1$, and for $C_n$ the highest short root is $\omega_2$. And similarly it can be checked that for $G_2$ the highest short root is $\omega_1$, and for $F_4$ the highest short root is $\omega_4$.

A table of these quasi-minuscule weights appears in https://books.google.com/books?id=Np7y-LVcwSwC&pg=PA221#v=onepage&q&f=false.