Progressive Web App in Chrome Custom Tab
Chrome Custom Tabs has the ability to show an "Add to Homescreen" menu item, at least in recent builds of Chrome.
I can confirm that manually selecting the "Add to Homescreen" menu item will make use of the web app manifest data when creating the homescreen item.
However, I don't believe that the logic for automatically triggering the Web App Install Banner is enabled for page views with Chrome Custom Tab at this time—I experimented a bit and never got the banner to appear, at least.