As outlined in #2785, let's extend the block settings menu to offer cut + copy options, where the block content is added to the clipboard.

Cases where Gutenberg is adding blocks to the clipboard:

https://github.com/WordPress/gutenberg/blob/b95e865de32ee62d722be3e3e71b334a4b639032/packages/block-editor/src/components/copy-handler/index.js#L26-L57

https://github.com/WordPress/gutenberg/blob/b95e865de32ee62d722be3e3e71b334a4b639032/packages/components/src/clipboard-button/index.js#L27-L67

Pasting would be handled by the RichText component which we use in all the text-based blocks and has some complex logic for handling the clipboard data properly:

https://github.com/WordPress/gutenberg/blob/03121c5e59c63d4c79a293f71bbb7b9613fd311c/packages/rich-text/src/component/index.js#L187-L260

We probably need to copy most of that component's logic.