Here is the complete solution, without needing to make manual changes after every build of the book.
(Note: this stops code blocks from having the copy to clipboard button, and adds it to the blockquote blocks.)
Add the following CSS to the style.css file:
blockquote {
position: relative;
}
blockquote:hover > .copy-to-clipboard-button {
visibility: visible;
}
Add the following lines to the end of the _build.sh file
sed -e "s/div.sourceCode/blockquote/" ./_book/libs/gitbook-2.6.7/js/plugin-clipboard.js > ./_book/libs/gitbook-2.6.7/js/plugin-clipboard_new.js
mv ./_book/libs/gitbook-2.6.7/js/plugin-clipboard_new.js ./_book/libs/gitbook-2.6.7/js/plugin-clipboard.js