I can't think of any built-in option for this. It's not listed in the help article: https://support.mozilla.org/en-US/kb/mouse-shortcuts-perform-common-tasks#w_links

Mozilla has a product suggestion site at https://connect.mozilla.org/. I think if you have time, it would be a good idea to post this request there to get into the current feedback stream. Use the "Ideas" section of the site.

I've even added a toolbar button to open a new private browsing window, then dragged and dropped a link onto the button, but that gets a circle with slash icon and nothing more.

When it comes to dragging, if you have a private window open, you can drag and hold the link over its button on the taskbar and when it comes up, you can drop the link into the current tab to replace that page, or drop it all the way up on the tabs bar to open the link in a new tab. You normally need to hold down the shift key to open the link in a new active tab, otherwise it opens in a background tab.

Right-click the link then press the P key.

Thanks @zeroknight, your suggestion works well enough to satisfy my need. I laughed when I realized the the P key is underlined in the context menu, yet I never thought to press it.

Working from your suggestion, I also found that, if the cursor isn't already in a text box, one can hold down/press P, then right-click the link, and the link will be opened in a Private Browsing window. Again, the cursor cannot be in a text box for this to work.

Thank you, too, @jscher2000. I will submit a feature request.