<< The problem seems to lie mostly at the straight grain edge of the circles. As I go around the bias edge, I struggle, but not half as bad as at the straight grain edge.

Greetings,You are absolutely right, that this is a grainline problem. When sewing two circles together, whether of the same type of material or not, you need to make sure that the grain of one fabric is cross-grain to the other fabric. In other words, Circle 1's grain should be horizontal to you, and Circle 2's fabric should be laying on top of Circle 1, in the vertical position,that way you will always have the stability of the grain of at least one of  the fabrics working for you. And keep pinning.  Happy Stitching,