I have finally found a way to manually refresh the favicons in Safari on the iPad! I set the date on my iPad to about a month in the future, then went to Safari settings and cleared the website data for the site with the misassigned favicon. After revisiting the site, the correct favicon was displayed. I then set the date on my iPad back to normal.
Just deleting the website data alone (or Safari’s entire cache) didn’t fix the issue for me—I needed to also set the date to the future. I presume there’s a favicon cache which doesn’t get deleted when clearing the browser cache, and that the favicon cache only gets refreshed after a timeout. Setting the date to the future presumably forces this.
Apple really shows how little they care for the web, with bugs like this… let alone that favicons get misassigned in the first place!