These days I want to take more control over my content, so I try to wrap my head around bridgy to tweet from my website instead of pulling data back from Twitter.
It works by wrapping your HTML in a .h-entry .e-content and placing a link to https://brid.gy/publish/twitter on the same pash (link text can be empty).
Then send a webmention from that website to https://brid.gy/publish/twitter. In case you receive a 400, check whether you enabled publishing in Brid.gy.