I do think your argument argument here is a little weak because you seem to be equating tricking people into your sanctuary where you will be permitted to murder them is a little bit different from running a business and not allowing certain people into your business.

It's reducto ad absurdem, but done to illustrate a clear-cut case where "Open House" -> "You're Trespassing!" creates a clear moral hazard. The problem remains. It's not merely "not letting someone in", but establishing an open invitation and then selectively kicking people out due to no action on their own part. It amounts to assault.

There's a very clear-cut way to manage a business that is invitation only. Run the business as invitation-only. The idea that business owners need a new suite of powers, endowed by the state, to call in police or eject invitees violently without any recognizable cause is merely an endorsement of a privatized police state. When you can be threatened or assaulted or arrested for sitting at an open lunch counter, you're living in the kind of thuggish society Libertarians allegedly oppose.

In any case I do agree with the spirit of what you're saying, I just think you're trying to make your position more appealing to libertarians by debasing it with a weird analogy

When in Rome...

There's more to the story, of course. Private businesses benefit from a host of public services (police, transport, tax incentives and state-subsidized loans, etc) such that no public business can truly claim economic independence. And when you're on the taxpayer dole, suggesting that you can use taxpayer funded services to oust peaceful customers without so much as a warning is 'effed up. But the standard Lib reply is "Abolish the state, problem solved!" So appealing to reality doesn't get you far when the goalposts are out in an idealized Libtopian.