Fix a scheme $S$, a group scheme $G/S$ (let us say smooth, maybe even affine with some finiteness conditions if you like), and suppose I have some other $S$-scheme $P$ with a right $G$-action. We want to investigate the question of whether $P$ is a $G$-torsor (in the fppf topology).

If I understand correctly, then since $G$ is smooth it is equivalent for $P$ to be a torsor in the etale topology, which is a priori a relatively strong condition. On the other hand, an a priori weaker statement is that it be a torsor in the fpqc topology.

My question is: suppose I can find an fpqc (but not finitely presented) cover $\{S_i \rightarrow S\}$ with $P\times_S S_i \rightarrow S_i$ isomorphic to the trivial $G\times_S S_i$-torsor. Are there circumstances under which I may conclude that $P$ is in fact an fppf torsor?

One (perhaps silly) example is if I have $S$ equal to the spectrum of some small field $k$ with an interesting Galois group, and I am able to find some huge complicated transcendental extension $K/k$ over which $P$ becomes a trivial torsor. Under what conditions does this imply the existence of a finite separable extension over which $P$ becomes a trivial torsor?