Sorry you’re finding this so frustrating 
I think you’ve been looking at the obvious things - cards, and then web intents. Unfortunately right now, there’s no way to add an image to a Tweet via a web intent / Tweet button on the web. In the case of cards, you would need every image to have a unique URL and web page with embedded cards metadata.
The only method I can think of that would enable you to do this would be to have users of your website sign-in to a Twitter app you would have to build, and then use the media/upload and statuses/update API endpoints to post the image on their behalf. We have a large video upload sample app written in Python which demonstrates the general patten for how to do this, but it would need to be modified to work in a web context.