It does not need to interact with other Mastodon instances, but users on the local server should still be able to follow accounts on other Mastodon servers.

This sentence is in conflict with itself, follows are “interaction”

littlefedi is out as of coupla days ago. still in early stage but should cover your use case. local binary, world comms via “lighthouses” don’t need domain, vps, etc. there’s something strange afoot wrt the author’s willingness to be forthcoming about the project’s provenance. I’d stay clear for the time being.

You can use a subdomain, and I’m pretty sure you can get a subdomain for free. It won’t be pretty but it will do what you want.

Have a look at GoToSocial.

Adding to why it has to be reachable: when a local user follows someone on another server, that server pushes new posts to your server’s inbox, so the remote side has to be able to resolve your domain and connect over HTTPS. A LAN-only box can’t receive that.

The least painful compromise is a cheap domain (or a subdomain you control) plus a tunnel or a small VPS acting as reverse proxy, so nothing on your home network is exposed directly. Then close signups so only your household can register. If Mastodon feels heavy for a handful of users, GoToSocial is much lighter on RAM for that use case, though you’d use a separate client app with it.

One thing to decide up front: the domain is baked into every account and can’t be changed later without starting over.

entirely within LAN

Wat? LOL