I've used white Warm and Natural cotton batting. Maybe it's called Warm and White?
I've also used "thicker" white cotton that's less see-throughy for the white.
Some white on white prints also seem to limit shadowing.
I think a loose lining would just be one more thing to rumple. Maybe a super-thin fusible?
Pressing 'to the dark side" limits some shadowing. When sewing a dark to a light/white , I have the white stick out a couple of threads from the dark, so after the piece is pressed, the dark side is covered.