I recommend using Lighthouse, which is an excellent tool for auditing your web app with respect to different categories, including PWA properties. It is also available in Google Chrome in the developer tools (Ctrl-Shift-I / Cmd-Opt-I and select tab Lighthouse).

How do I trigger a PWA audit in Lighthouse (in Google Chrome)?

Choose the category Progressive Web App and then click on Generate report.

Example report

In the example below, you can see that there are some issues present. But the reason, why you don't see the installation prompt is the service worker. Apparently, it isn't registered as expected.

Web app manifest or service worker do not meet the installability requirements

I faced a similar issue. I came to the conclusion this issue could be due to the browser checking for a valid manifest before your dynamically-created manifest has been inserted into the DOM.

If the browser thinks there is no valid manifest available for PWA installation, then the install prompt will not show, which is stated clearly enough in the documentations. However, debugging this issue is rather confusing, as tools, such as Lighthouse in the Google Chrome Audit DevTool tab will say that the app is installable and everything is fine and dandy...

It could be the case that the browser only checks for a valid manifest once during page load - unfortunately I can't find any firm details anywhere about this.

I was able to solve this issue by ensuring the html document's initial render has a valid manifest for PWA-installation, containing some default values (in my case, the values were based off of window.location.pathname). Then later on, when I have all the appropriate data available, I can create my desired manifest and replace it in the DOM. Then, when is ready to install the PWA, the browser uses the data from the desired manifest.