The question relates to a general polygon, but I guess a single triangle is a reasonable reduction.
I can formulate the problem as follows: Given three points $v_1,v_2,v_3$ on the unit sphere, and three quaternions $q_{12}, q_{23}, q_{31}$ that transform the respective points to each other, what is the signed area enclosed by the interpolated curve? Assume it is supposed to be right-handed positive for an outward normal to the sphere.
The transformations are not necessarily geodetic (great circles), which mean we don't have that $q_{ij}=v_i^{-1}v_j$ (assuming $v$ as imaginary quaternions). This rules out the simple Girard formula for spherical excess (or does it?).