Thought that wasn’t possible, but it actually is. Here‘s two approaches:

Change the apple-touch-icon urls

I made this little shortcut that does it for you, you just need a url to the image and run the shortcut on the website you want to add.

(If you want to use the space mouse/lemming icon I made, here’s a high res image url: https://i.imgur.com/kUxXoIA.png)

Or if you want to do it yourself:

You can manually change the apple-touch-icon urls before adding the web clip to your home screen. Either by connecting your device to a Mac and doing the changes via web inspector there, or directly on device by using a safari extension such as this one that lets you run JavaScript.

You can change the url to any image url you want to use.

iOS Profile

Add the web clip via profile by running this shortcut. You don’t need to upload the image anywhere, but it means you have the profile laying around.