Thanks for that suggestion. I found a couple of containerized multiplayer "servers": https://github.com/cavazos-apps/stardew-multiplayer-docker and https://github.com/JunimoHost/junimohost-stardew-server that seem to (as best I can tell) both run a game client as the server. So it sounds like a dummy player seems to be the way to go.
Haven't tried either of those yet though. I assume the JunimoHost one, at least, won't work with the current version of Stardew.
I've found a couple mods to change timescale that would let me have 1 second = 1 second, but nothing so far that would let me match the time of day properly. Now I'm considering writing a mod for this myself. I just need to decide if all of this is actually worth my time...