You can have a website with login that is outside of the OSA, it's only the user to user communications that drags it in. For example Ofcom explicitly cites review sites as an example they consider do not count - so amazon is outside the OSA, but you log into that. All you need to do to keep the "forum" online and outside the OSA is remove any mechanism for one user to talk to another directly.
Now depending on what you want the "end goal" to be there could be a few options. If you'd like to keep it as it is where a guest can only view 'X' posts/pages/whatever before registering then fine - just leave that add-on enabled. You'd then just need to ensure that the permissions prevent your general members from posting threads/posts/profile posts/tags/reactions/PMs/etc. That will all slightly depend on how you have your groups set up. You may find you need to add additional enabling permissions to your admin type groups (or directly to those users) if they exist so they can still do a bit tidying, etc. Your existing users would then be able to log in, read anything already posted - change their details, etc, but NOT be able to post anything. Guests such as myself would be able to read a few pages then be prompted to register. I could then register and presumably read anything already posted, tweak my account, but NOT be able to post and interact with any of the other users. Obviously it'd also make sense to add a notice somewhere saying what has been done and why!
A site like that would be outside of the OSA from my understanding of it.
A word of caution with archiving the site - if you are archiving it as your user - who presumably has admin permissions then the crawler might well archive stuff that shouldn't be shown to the public - so whilst that archive might be good for your own personal consumption/reference it may not be suitable to have online. So if you have given the crawler a login you might want to make sure it's a "general user" who can just see most stuff, but can't go off and see things like reports, user details, etc.
Best of luck getting it to a state you are comfortable with.