Absolutely irreducible representations of affine group schemes of finite type over a field

Suppose $G$ is an affine group scheme over an algebraically closed field $k$, and $V$ is a finite dimensional representation of $G$. Let $K$ be an extension of $k$, and assume that $V_K$ is reducible as a representation of $G_K$. Then I claim that $V$ is reducible as a representation of $G$.

Let $r$ be an integer with $0 < r < \dim V$, such that $V_K$ has a $G_K$-invariant $r$-dimensional sub-vector space. Let $\mathbb{G}(r,V)$ the Grassmannian of $r$-dimensional vector spaces of $V$, with its induced action of $G$. Since formation of scheme-theoretic fixed point loci commutes extension of base field, the fix point locus $\mathbb{G}(r,V)^G$ has a $K$-valued point. This means that is non-empty; since it is of finite type, and $k$ is algebraically closed, this means that $\mathbb{G}(r,V)^G$ has a $k$-valued point, so $V$ is reducible.

The classical viewpoint is captured well in the 1962 book Representation Theory of Finite Groups and Associative Algebras by Curtis and Reiner (Wiley), Corollary 29.15. This of course doesn't directly answer the question about non-reduced representations of affine group schemes. But I suspect this generalizes. See for example II, $\S2$ of the 1970 book by Demazure and Gabriel, Groupes Algebriques (Masson, Paris). This book is written in the language of affine group schemes.

(By the way, my comment on the question here was being edited when I was interrupted by a phone call and didn't finish the editing.)