I can't suggest anything, I'm afraid—I haven't worked with Windows and Linux targets using React Native. But it appears that there are React Native rendering targets for Windows and Linux (https://github.com/Microsoft/react-native-windows and https://github.com/status-im/react-native-desktop). I'd probably start by looking there.

And of course, because React Native can build to web, one could ship Electron apps for platforms with lower priority for your business.

Yes, at least officially for Windows. The official fork: https://github.com/microsoft/react-native-windows