check if user has already installed PWA to homescreen on Chrome?

To answer original question. With latest versions of Chrome you can use window.navigator.getInstalledRelatedApps(). It returns a promise with an array of installed apps that your web app specifies as related in the manifest.json. To enable this to work you need to add related_applications field to manifest.json

  "related_applications": [{
    "platform": "webapp",
    "url": "https://app.example.com/manifest.json"
  }]

And then you can use it like:

//check if browser version supports the api
if ('getInstalledRelatedApps' in window.navigator) {
  const relatedApps = await navigator.getInstalledRelatedApps();
  relatedApps.forEach((app) => {
    //if your PWA exists in the array it is installed
    console.log(app.platform, app.url);
  });
}

Source: API docs

Now you can display some elements depending if your app is installed. E.g: you can display "Open app" button and redirect user to PWA. But remember to disable it when the user is already in the app using @Mathias's answer and checking (display-mode: standalone)

However, regarding your use case. You should display install button only when beforeinstallprompt is intercepted. Browser does not fire this event if the PWA is already installed on the device. And when prompt is fired and choiceResult.outcome === 'accepted' you hide the button again.

Perhaps, don't show the button until you intercept the automatic pop-up?

or
In your code, check to see if the window is standalone
If it is, you need not show the button

if (window.matchMedia('(display-mode: standalone)').matches) {  
    // do things here  
    // set a variable to be used when calling something  
    // e.g. call Google Analytics to track standalone use   
}  

My example tester here
https://a2hs.glitch.me

Source code for my tester
https://github.com/ng-chicago/AddToHomeScreen