If you just want an *image* of a Tweet, you can use this URl:
Replace the ID at the end with the one you want, and use a screenshot tool to grab the post.
Again, no API key needed. Just raw access to Tweets and their contents.
Anyway. Use this knowledge to remove all the Twitter embeds on your site and replace them with static content.
I've written some helper scripts in Python at https://github.com/edent/Tweet2Embed
If you have a fediverse account, you can quote this note from your own instance. Search https://mastodon.social/users/Edent/statuses/114229757525633180 on your instance and quote it. (Note that quoting is not supported in Mastodon.)