A dumb idea that probably doesn't have an implementation: Set Thunderbird to play a sound on mail arrival, but have the sound file actually be a pipe that when read from also pushes a system notification. This is kind of like how randomised .signature files were often set up in the old days.
Other alternatives: 1: There might be a purely mail checker out there that can log into mail servers to see if there's new mail there but not be able to read or download it.
2: Run your own mail server that pulls mail from other servers. Then it's "merely" a matter of checking for file update times on your own machine. Ancient tools like xbiff were designed for this.