That's a really strange issue. From what you've described the only thing I can think of is you're using tno.SendQuickly for most of your RFCs and UDP is failing (though SendQuickly should fall back to TCP if UDP is unavailable...). TNManager.Create uses TCP so that explains why some actions work for these invisible clients.
I would check to see if your port for UDP is opened (both the server port and the invisible client's port). You can find web-based port scanners online. It's still really strange that it's not falling back to TCP. Maybe you've got some insane packet loss when using UDP?
I'd load up wireshark and inspect packet flow. I'd also set up some breakpoints in your project and step through the important bits.
That's all the help I can provide without more info. Good luck!

I'll check back periodically for updates.