You've essentially answered your own question. Let $G$ be the units of a division algebra of dimension $n^2$. Then $G$ is an inner form of a general linear group so the Galois action on the root datum will be trivial. But the weight corresponding to the canonical $n$-dimensional representation of $G$ over the alg closure descends to an $n$-dimensional representation over $k$ iff $G$ is split.
EDIT: Bcnrd raises the issue that the questioner says "everything has an action of Galois" without saying what this action is. My answer implicitly assumes it is the action called $\mu_G$ in Corvallis (which has the property that it depends only on $G$ and not on $T$) and Bcnrd raises the issue that it could be the the "naive" action (which depends on the arithmetic of $T$). I do not know which action the OP means, and the validity of this answer is contingent upon my guess being the right one. UPDATE: BCnrd tells me that in the paper in question, the action seems to be the one I'm calling $\mu_G$ so this answer is probably OK, but it does leave open the question as to what happens if one uses the naive action.