Well, then, it shouldn't be a problem to write a service that automates the capture of screenshots of tweets posted by a particular account or on a particular hashtag. That doesn't even require access to the Twitter API, or, for that matter, having a Twitter account.
1. Tweets wouldn't be stored. Screenshots of tweets would be stored.
2. Admittedly, that would be the most inconvenient part of the process. This would require regular scraping to validate what's been deleted and what hasn't.
