I think what you want is a private forum. Someone will have to pony up some light server costs, but as long as you aren't using a lot of bandwidth — you could disable image uploads, and make your users upload to Imgur and the like instead — it shouldn't be much.

I remember seeing a private forum. It was for the movie Boys Don't Cry — or rather, the trans man the movie was about. He was a real guy, and he was murdered for being trans. There's a forum out there (or, there was 20 years ago) and all you can see is the name. You need a password to see anything else. Don't have that? You're probably not getting in. It's a place for his friends and family, and probably some trusted friends they've made along the way.

I ran a forum once, about 15 years ago. I think I paid $10 or $20 a year for the dot-com, the URL. And I was paying a company called Invision something like $10 a month for them to host their forum on their server. I had admin access, I could do anything I wanted with it, but I had bandwidth limitations, and I had space limitations. It was fun for a bit. I ran it for almost two years. I wanted out, and a guy on the forum wanted to take it over. I said he could even keep the name, but I wasn't going to keep paying for the address. I didn't sell it, I just transferred ownership over. I never got charged again. I didn't keep visiting it. I heard he shut it down a few months later. It's not for everyone. If you're not passionate about running a forum, get someone else to do it for you, and just have them maintain it. You don't even have to be a mod. You let people apply and you tell your guy who you want to be a mod. He runs it from the shadows, only there to do upgrades. Maybe he posts the rules, too. But doesn't participate in conversations. Maybe once in a blue moon you see him, but he's not what you'd call a forum regular. Or he's a snarky arsehole, I've seen admins like that.

You really can't have a super private Lemmy comm. Not sure about instance. With the comm, you can make it so only approved people can post, but I don't think you can hide it from everyone by default. That's not what Lemmy is for, and we're too small anyway (though, I don't mean to speak for the people in charge). I think the forum thing might be your best bet.