I’ve ended up building my own version of the button with the right destination URL so that the user can automatically follow. So it’s faking the behaviour of the button generated by Twitter, and I have full control over it to e.g. add accurate tracking.
It’s a pity I haven’t been able to find out what the real problem is, but at the end the important thing is finding working solutions, even if they are not the official ones! 
Thanks everybody for the help!