Unfortunately, I do not have a solution for that. I don't know how QuickConnect / DSM authentication might interfere with each other. AFAIK,
FileBot Node is doing that the DSM documentation specifies. I'd be open to code changes though if I knew what to change.

I would recommend OpenVPN if you need reliable remote access to your local network, or maybe something like
Tailscale. That way everything will work as if you're on the local network.

QuickConnect is made specifically to connect to the DSM WebUI and generally doesn't work with any 3rd party applications
(e.g. Plex, docker containers, etc) even though FileBot Node might work to some degree by nature of piggybacking on the nginx server that serves the DSM WebUI.
EDIT:

You do go through the DSM WebUI login before opening FileBot Node, right? If you happen to be opening the FileBot Node
iframe directly - bypassing DSM login - then that won't work.