Since the new Clipboard API, this is easy in browsers that support it:
javascript: navigator.clipboard.writeText('some text from somewhere');null;
Caveat: Any alerts or prompts in your bookmarklet will cause the document to lose focus, and the clipboard API will become unavailable. In that case you need to move focus back into the document before any subsequent clipboard operations will work:
const oldFocus = document.activeElement;
/* do popup stuff here */
oldFocus.focus();
/* do clipboard stuff here */