I’m experiencing the same issue and I tried messing with some options in the exporter. Unfortunately, swapping the axis of primary and secondary bones didn’t work for me.
I found that when exporting fbx animations, if I turn off ‘simplify’ or make it smaller values like 0.1 it would significantly improve the result.
And here is the comparison:
(linking my gifs from external since I can’t post more than one image)

(Broken: FBX exported with ‘simplify’ being 1.0)

(Fixed: FBX exported with ‘simplify’ being 0.1)
Decreasing simplify would result in larger files in size but it’s a smaller issue comparing to the animation quality.
