You can just call bpy.ops.image.save_sequence() within the Image Editor.
Override the context
If you would like to call the operator from another area, you can override the context:
C = bpy.context
for area in C.screen.areas:
if area.type == 'IMAGE_EDITOR':
override = C.copy()
override['area'] = area
Thank you for posting this question. I encountered a similar problem too, here is what I found:
I don't have any modifier on my model but the symmetry texture paint is still off. So I apply All Transforms to my models' parts in object mode by clicking ctrl A. Then the symmetry texture painting work like a charm.