Although the copper IUD is non-hormonal... It definitely has some effect on hormones. The copper disrupts the estrogen/progesterone balance, which could be a similar problem with the Mirena. And vice versa... the estrogen/progesterone imbalance has an effect on the copper/zinc ratio.
If you read the book: Why am I always so tired? by Ann Louise Gittleman she states that there is...
"...a close relationship between copper and estrogen levels in the body, and between zinc and progesterone."