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.

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.

Edit: If I interpret the description of the projective bundle in the question correctly, the $\mathbb{P}^1$-bundle is exactly the variety cut out by the vanishing of the norm on the projectivization of the quaternions of trace zero. The same can be done for the octonion algebra above: the equation $(sY^2-tZ^2+stT^2)-r(U^2-sV^2-tW^2+stS^2)=0$ should define a hypersurface in $\mathbb{P}^6_A$, defined by vanishing of the norm on the projectivization of the octonions of trace zero. If I'm not messing up something, this should be a homogeneous space bundle with fibers $G_2/P_1$ which is not locally trivial in the Zariski topology. I think this is the appropriate $G_2$-analogue of the $\mathbb{P}^1$-bundle in the question.