Hi,
My doctor is really cool and down to earth and I trust what she says - she told me that if there is any chance at all for anyone to get a specific side effect they have to list it by law, but actually it is more likely it will increase your milk supply a little as your body adjusts to the hormones. I think I got a Mirena in around my 6 week checkup, and never had any problems with milk supply. Also, I never had a period while I had it, although I was exclusively breastfeeding also. I think the first day or two I had a little spotting and cramping, but then it was like nothing at all. The one thing I will warn you about is the string that hangs down from it so they can pull it back out when you want it removed is very stiff and can take a year + to "soften" (or that is what they told me). I had mine trimmed a few times, and even removed it and tried a second one, but both times the string scraped against my husband and was very uncomfortable for him, so I ended up going back on the pill.
I know you didn't ask about the pill ... but if you don't do the IUD I assume you will choose something else. If you are breastfeeding they will put you on the progesterone only pill, which is what I am on again right now. I don't really like it, since it doesn't have the estrogen and the "off" 4th week it doesn't make your cycle normal, and for me anyhow, it doesn't help with cramps and has been giving me a few zits here and there. But I am on it because I had been on YAZ for a few months and realized after having 2 migraines in 8 days, and milder headaches on the other days (I don't usually get headaches) that maybe they were being caused by the hormones. Sure enough, the day after I switched pills the headaches were gone. So it was either the increased levels of estrogen or the variation of synthetic progesterone that is in them. I never knew before that there are several different kinds of synthetic progesterone that are used in different pills.
Anyhow, this is more information than you asked for, but I hope it helps. If it wasn't for the string I would totally have a Mirena IUD, I loved it, and I know that not all people have that problem. Good luck!