As such, this distribution would need to be able to handle running for weeks on end without a reboot.
So, it has to be something stable (i.e. receive little to no updates) that's capable of updating without requiring a reboot. That makes any stable distro a candidate. As such, choose either:
- Debian or something based on it
- Gentoo (stable branch)
- Nixos (stable channel)
- openSUSE Leap
- Ubuntu or something based on it