I have good news - after some tests it seems even session on cloudflare access will expire somehow Home Assistant app remember notification settings and notifications still works.
You can do this in another way by using Cloudflare Warp Tunnel (which then assures the source proxy server so JWT then isn’t needed) - and use the following header to get the authed user email: username_header: cf-access-authenticated-user-email
Another option is to use HA Proxy to parse the JWT and insert the username into an X-Header as well and again use header auth for HA Proxy.
To get the app to work behind CF Access I bypassed /api however SSO still works fine.