Sorry, Matt, I’m just seeing this now. Not sure how I missed it originally unless there was such a huge bunch of responses on the site that your comment got scrolled further back than I managed to catch up with. It’s been a busy month and change.
In any case, replies via Twitter (and similarly Google+, Facebook, Instagram, Flickr, and even GitHub) are done courtesy of a lot of the IndieWeb technology underpinning my site. I’ve documented the Twitter specific portion in the past, but the others are done very similarly once you’ve got Twitter working: http://boffosocko.com/2017/04/15/mentions-from-twitter-to-my-website/
If you want to extend the functionality and UI a bit you might also try this on for size: http://boffosocko.com/2017/12/24/adding-simple-twitter-response-buttons-to-wordpress-posts/
Apologies again for not having seen this sooner.