Running an Ionic app as a PWA on Firebase hosting - google-app-engine

Question

Background
We have a large Ionic v1 app that runs on Android (by obtaining it from the Google Play Store) and also on our development machines via "ionic serve".
The app uses a Google App Engine (GAE) website as a backend server.
The server maintains sessions for each user by means of a cookie. We don't store much data in the session, but we need to securely identify the user making each request. When the app runs successfully, the GAE server code creates a cookie that contains the session ID and sends it to the Ionic client code when responding to each HTTP request.
Note:
That the Ionic code does not access the cookie in any way. It is only
necessary that the same cookie be sent back to the GAE server with each
subsequent request so that the GAE code recognizes the user.
The goal
We would like to serve the Ionic code by use of Firebase Hosting. We can in fact do so in both of two modes:
Keeping the Ionic code on our dev machine, running "firebase serve", and going to "localhost:5000" on the browser
Deploying the Ionic code to the Firebase host and going to "xxxx.firebaseapp.com" on the browser
Everything works! Uh, except for one little thing, which we've been trying to solve for weeks...
The problem
The cookie used by the GAE code to manage session continuity, and sent in responses to HTTP requests generated by the GAE code, does not come back in the next request from the Ionic app running on Firebase. So the GAE app always responds as though the user is not yet logged in.
In fact, further testing shows that the session cookie sent in responses to HTTP requests does not even get set in the browser (so of course it's not sent back to the GAE code with the next HTTP request). The GAE code on the backend server always responds as if this is the first HTTP request of a session.
What we've solved already
The problem is not the fact that Ionic does not support cookies. We know this is not the problem because the app runs fine as an Android app and also via "ionic serve". In both cases, the GAE backend is able to maintain sessions using a cookie to store the session ID from one request to the next.
The problem does not get solved by using "memcache" instead of cookies for GAE session support, because even if you use memcache, you still need the cookie for the session ID. If you wish, you can go with the default and let GAE session support use cookies; in that case, it will use the same cookie for both the session ID and any other session data.
The problem does not get solved by using "__session" as the name of the cookie. Firebase does in fact support using such a cookie name, but apparently only in the context of running Firebase Hosting with Cloud Functions. Cloud Functions are for running backend code, not client code that the user interacts with. We could see no way to make an Ionic app run as a Cloud Function. And without Cloud Functions, the "__session" cookie set by the GAE backend apparently gets stripped by the browser client running the app, along with all other cookies.
Adding "Access-Control-Allow-Origin/-Credentials/-Methods/-Headers" headers to the GAE-code generated response, and setting crossDomain: true xhrFields: { withCredentials: true } on the client side, does not improve the situation. Still, no cookie from the GAE code gets set on the browser.
Any help will be appreciated.