Install High Sierra using Screen Sharing
I've tested this by screen sharing a VM and it seems to work great!

Using macOS screen sharing app, connect as an administrator and be the only user logged in to the machine at the moment. If, on connecting, you get a message that another user is logged in and whether you want to share session or create a new session, cancel and log in as the correct user. The update will proceed correctly regardless of whether you do this, however you won't be able to see the progress of the update, as other sessions are disconnected immediately.

I'm not aware of any macOS update which has ever required input outside a user account. The setup assistant (‘Sign in to iCloud’, etc) happens after you log in to your user account again, which is usual screen sharing.

Upon starting the update, the user is logged out but the progress bar is still a standard macOS application. Screen sharing continues to work fine at this point.

Once the ‘progress bar’ stage of the installation is complete, the machine reboots, and screen sharing is disconnected. macOS screen sharing app shows ‘reconnecting’ and eventually the connection is resumed. If your Mac takes a while to reboot, screen sharing may time out, so wait a while and manually reconnect.

If you have Little Snitch installed, it should be fine, but might also be worth choosing ‘Silent Mode — Allow Connections’ just in case for the duration of the update.