is it a suggestion or a polite order/command? I think the best way to say this as an order is: "I'm going to have to ask you to leave"

In English the word, invite, is too welcoming for this type of situation. If you want to be polite and business-like, you could say, "I suggest that you leave" or "I think it is best if you leave ". Saying, "I request that you leave immediately" sounds more intense and less professional, as does "I'd like you to leave now".

English has the same expression. He was invited to leave. (But it’s rather formal and a little old-fashioned. ) Teacher. “John, please leave the room” Other students talking about it later “Did you hear that John was asked to leave?” “Did you hear that John was kicked out of class?”

I would say "Please go home", or suggest "Maybe it's time for you to leave/ Perhaps there's somewhere else you need to be". There's a lot of different ways. If it's an indirect suggestion, normally people understand that they should really just leave and not say anything else.

Thank you very much Bria and Peter.