I would be curious about the question in general, and now that you've given your counterexample in the function field case, I think something similar will work in general. For a division algebra of degree $n^2$, there should be $S_n$ number fields $K$ which split it, and those will give counterexamples - using the Galois action on lattices viewpoint, an extension will split a torus only if it contains a large enough (I'm being vague here, but you probably see what I mean) galois subextension. As $K$ is an $S_n$ field, it's not going to have this property.

I'm interested in the question over number fields, so I'm happy to assume that my extensions are separable. Thanks for your answer - I worked a messy answer out for the quaternions, and it's good to have it confirmed for CSAs.