You could try slerping between A and B based on their relative weights, then slerping the result to C based on its weight.
For instance, in your example, A and B have a total weight of 90%. Use this to normalize their weights to get A' = 66.7% and B' = 33.3%, so slerp from A to B by 33.3% (or B to A by 66.7%, equivalently). Then slerp from that result to C by 10%.
For ordinary linear interpolation this is equivalent to adding up points weighted by barycentric coordinates (i.e. the usual method of interpolation for triangles). As you would expect, for linear interpolation it doesn't matter what order the points are in - you get the same answer regardless.
In the case of slerping, I don't know if you get the same answer regardless of the order of points, but you could try it and see.
For more than 3 points, you could extend this method to incorporate points one by one, using weights normalized by the sum of weights of all the points you've incorporated so far.