Maybe you have to have a url to the image, but you could just upload it to imgur und copy the link there. You can replace the icon url by searching for "apple-touch-icon" and changing the href property everywhere you see it to the image url.
On iOS, you can use the open url scheme in Shortcuts and assign any icon you’d like when you add the shortcut to your Home Screen. Not sure about Android though.
Edit: Yeah not recommended for a web app though but it does give you a custom icon.