I believe that to do the button you will need to do a lot of advanced work that is just troublesome, rather have a explanation for the user or prompt them to open the application twice. This has enabled me to get a popup that askes the user if they want to install the app after using the app link twice.
Please look into using something like this, otherwise you will have to find the piece of code that controls the prompt or you will need to write your own HTML snippet that will action this function for you in the browser.
You may consider to use the PWA Builder tool, that can convert your PWA in a format able to be published to the Stores.
From the stores, your client will be able to install the app as usual.