How to prevent a URL in an email I send from being auto-linked by the recipient's email client?

You can edit the HTML of the link before you send it. Just click the Edit Hyperlink... showed in your image there and then add the <z></z> tag within the hyperlink, as outlined in the SO answer you showed.

This has the advantage of showing up like a link, but attempting to click it does nothing. However, you are not prevented from selecting it to copy/paste.

The z isn't necessary; any invalid HTML tag will do. You could replace the whole link with something invalid, but then it would still try to open it in a browser. For some reason, having the invalid tag in the middle prevents it from even trying to open it even though it still knows it's a hyperlink.

You can't prevent this (without somehow breaking the URL), because it's the recipient's client that displays the URL as a clickable link.