This question is motivated by my attempts to answer the question Invariants for the exceptional complex simple Lie algebra $F_4$ from the point of view of Tannaka reconstruction. This has led me to the following problem and I am asking for an example (or alternatively an argument that what I am asking for does not exist).
Let $K$ be a field and $G$ an algebraic group over $K$. I will use "tensor" as shorthand for "$K$-linear rigid symmetric monoidal". Then the category $Rep(G)$ of finite dimensional rational representations is a tensor category.
Then let $C$ be a tensor category and $\omega\colon C\rightarrow Rep(G)$ a faithful tensor functor. There are two properties I am asking for. First, that $G$ is the group of tensor automorphisms of the functor from $C$ to vector spaces over $K$ (given by composing $\omega$ with the forgetful functor). Second that $\omega$ is not full.
Note I am not assuming that $C$ is abelian. This is a standard condition. If this is assumed and $\omega$ is exact then Tannaka theory implies that $\omega$ is an equivalence. I also suspect (but can't prove) that if $G$ is reductive then $\omega$ is full.
I would be particularly interested in an example with $K$ of characteristic zero (in which case $K$ is perfect and $G$ is reduced).