Better Forum Software
There is a lot of free forum software, such as FluxBB, PunBB, MyBB and others. They support BBCode, and some software allows you to add BBCode if necessary.
You can also show who's online on the forums, edit posts, make locked topics more obvious, and many more helpful things.
Plus, Matma Rex has a mirror for the current forum so even if you do remove the current forum you can still read posts from it.
An important feature of the WarLight forum is that it integrates with WarLight – i.e., it allows you to use the same login, link to your profile, etc.
The forum systems you mention won’t do this out of the box, and the open source ones are licensed under the GPL which prevents me from adding these features to them.
If you want a specific feature added to the forum, you can create an idea for that feature. If you find one that’s licensed under something more permissive like Apache or MIT then you can suggest that, but when I set out to add the forum to WarLight I could not locate any.
Switching to a GPL-licensed forum system is not possible, so I’m closing this idea.