The Online Safety Act (OSA) defines criteria to determine size.
Size then determines level of regulatory impact.
And what you've just described, falls comfortably into what they define as small category.
That means, less obligations compared to medium/large, but you still have to be compliant.
OSA isn't aimed solely at "X, Reddit and their ilk" - otherwise they'd have allowed for exceptions based on user count. They haven't. Intentionally, as there's a load of smaller, rogue forums too.
It's aimed at all online platforms.
You can read more on here.
So while it's technically, as you say, my forum, the rules by which has to operate, is UK Law.
And trust me, putting content behind registered users doesn't prevent the necessity to also have to moderate it in accordance with OSA/risk assessment.