I had this same issue. Apple wouldn't help me, unfortunately. If I recall correctly, I had to disassociate iMessage from my AppleID completely, which seems to have worked. Although I still can't be sure.
I have a weird feeling the messages still aren't going through and a lot of my friends think I'm being an asshole.
Honestly, this is great for Twitter. I can't see them blocking this. I can either advertise my product on Facebook which necessitates leaving the site in order to buy my product, or I can use Twitter Ads with #AmazonCart and see usage on a case-by-case basis all while staying within the confines of Twitter's app.