I don't think the app needs to support both modes. When you go in a browser, it should just say, click here to install the app to your home screen. That's the easy answer, IMO.
But then you're increasing the barrier to entry to users. You're assuming the user cares enough about your app that they will sacrifice a spot on their home screen and will want to tap three buttons to get to it...
Users already go through even more steps when getting native apps through the App Store, so I don't think that'll be an issue.