@greypants So I don't actually have it altogether end-to-end, however, I can tell you it is definitely possible. When I originally posted this, I hadn't really used any of the tech, but had a 50,000 ft understanding of it all. Conceptually, I was correct.
I'm making a basic React Native app that acts as a shell to the web app. I have a pretty intro scene to welcome the user to my app. Next, then the user logs in via Single Sign On technology. I capture the auth cookie in the SSO response, ask for push notification permission on iOS, and then POST the info to a server REST end point the device token and the cookie for authorization, so I can associate the given device with the user. On Android, can make the POST request directly -- no need to ask for permissions.
After that, I take the issued auth cookie and navigate to a new scene containing a full-screen WebView, inject the cookie, and load the homepage. Then whenever I need to send notifications, my application servers will have some business logic to take care of that. Probably going to go with Google's Firebase Cloud Messaging service since it's free (but AWS offers very low prices too). I'm probably going to put the URL to load in the payload, to keep the notifications contextual to the in-app experience.