Cannot do a `composer install` even though my minimum stability is listed as dev
This may be incorrect per information in comments and because it contradicts Installing an alpha dev version with composer.

I think the issue here is that the beta6 is not ^1.0. If you updated the requirement to be "drupal/focal_point": "^1.0@beta", this would work fine.

Alternatively, you could call composer require drupal/focal_point with no version specified and it would install the beta6 version just fine. I believe this is where minimum stability is checked, but that would add a  require line in your composer.json different from what is specified in your question.