Considering you are already using Rust and looking at a static binary: why not package that inside the various distro-specific formats?

Sure, you won't get into the official repos that way, but do your users really care? Fedora users are pretty used to enabling Copr and RPMFusion, Ubuntu has its PPAs. It's not ideal, but it's good enough. Even an RPM/DEB via Github release would already be a good start.