Disclaimer: I'm in the same situation as you and I haven't had a go-ahead from Amazon with this method.
You could cloak all the links to Amazon on your website like this:
Link to Amazon Product: <a href="http://yoursite.com/amzn/11223344" target="_blank">Product with ASIN:11223344</a>
Your /amzn/[ASIN] folder would then do a 302 redirect to the amazon product page with your affiliate code.
If done like this, it will be impossible to tell whether the click came from your website or the above link posted on Twitter.
As I said, I'm not sure if Amazon approves this (in fact I believe they make all cloaking against their ToS).
I was considering doing this for SEO purposes, but risking profit for whatever SEO benefit this gives me seems a bit stupid. If you decide to do this and it works though, please let me know.
You can use https://yoast.com/cloak-affiliate-links/