I think a construction like the one for ${\rm PGL}_2$ in the question should work. The bundle in the question corresponds to the quaternion algebra over $\mathbb{C}[s^{\pm},t^{\pm}]$ whose norm form is $T^2-sX^2-tY^2+stZ^2$, i.e., the quaternion algebra where $i$ and $j$ are non-commuting square roots of $s$ and $t$, respectively.

Now we do the same thing for octonion algebras. The underlying ring is going to be $A=\mathbb{C}[s^{\pm},t^{\pm},r^{\pm}]$. Apply the Cayley--Dickson doubling construction to the above quaternion algebra, with parameter $r$. (This works not just over fields, definitions or locally ringed spaces can be found in papers of H.P. Petersson.) This produces an octonion algebra over $A$ whose norm form is the 3-fold Pfister form
$$
\langle 1,-s\rangle\otimes\langle 1,-t\rangle\otimes\langle 1,-r\rangle=
$$
$$=(X^2-sY^2-tZ^2+stT^2)-r(U^2-sV^2-tW^2+stS^2).
$$
This is a non-trivial 3-fold Pfister form over the function field $\mathbb{C}(s,t,r)$ and so gives a non-trivial decomposable element in Galois cohomology ${\rm H}^3(\mathbb{C}(s,t,r),\mathbb{Z}/2)$. (You can think of this topologically, it's the top cohomology of the 3-torus ${\rm Spec}A$.) In particular, the corresponding octonion algebra is going to be nonsplit over the function field. Then we can take the corresponding $G_2$-torsor (of local automorphisms of the octonion algebra); by the above this torsor cannot be Zariski-locally trivial.

An example of a torsor for ${\rm Spin}(7)$ or ${\rm Spin}(8)$ can be obtained by the natural change-of-structure-group, alternatively, these can be explicitly described in terms of the 3-fold Pfister form above. I guess the 3-fold Pfister form gives a family of smooth affine quadrics whose generic fiber is anisotropic, another example of a torsor which is not Zariski-locally trivial.