Thanks for all the great answers. Recently it appears that things have become more standardized, and there is a solid way of doing things in chrome at least that allows users to click and install the app even if they missed the prompt etc.

Pete LePage wrote a very clear article on web.dev called How to provide your own in-app install experience which details the process. The code snippets and process is taken directly from the article.

1. Listen for the beforeinstallprompt event.

let deferredPrompt;



window.addEventListener('beforeinstallprompt', (e) => {

  // Prevent the mini-infobar from appearing on mobile

  e.preventDefault();

  // Stash the event so it can be triggered later.

  deferredPrompt = e;

  // Update UI notify the user they can install the PWA

  showInstallPromotion();

});

2. Save the beforeinstallprompt event, so it can be used to trigger the install flow later.

buttonInstall.addEventListener('click', (e) => {

  // Hide the app provided install promotion

  hideMyInstallPromotion();

  // Show the install prompt

  deferredPrompt.prompt();

  // Wait for the user to respond to the prompt

  deferredPrompt.userChoice.then((choiceResult) => {

    if (choiceResult.outcome === 'accepted') {

      console.log('User accepted the install prompt');

    } else {

      console.log('User dismissed the install prompt');

    }

  });

});

3. Alert the user that your PWA is installable, and provide a button or other element to start the in-app installation flow.

For more details and other capabilities, see the full article.

I made a working demo of a this process implemented in React. (source code)

Also, for a slightly different approach, you can see how the maker(s) of Snapdrop implemented an install button in the source code.