There was a similar issue.
We decided to simply periodically (every minute) upload accounts from the main site (Symfony 2) to the forum database (FluxBB). In the latter, they only changed the password encryption algorithm (a couple of lines).
The problem needed to be solved quickly, this solution in our case turned out to be optimal in terms of time / quality ratio.

All large forum engines have an API through which you can register / authorize users. Why doesn't it fit?

Probably few people use Symfony2 in conjunction with forum engines, or the task is very specific. In fact, it is difficult to solve it fairly evenly.
The easiest way to solve this problem is to use events and write a service for authorization on the forum.
Upon successful registration and / or registration, an event is called (I don’t remember exactly what and how, but a couple of months ago I had to implement some specific action for successful registration / authorization, something like notifications to the administrator). I made based on FosUserBundle with a few tweaks. Out of the box there is everything you need to solve your problem.
You can implement this without any bundle, arrange the library with the API as a service - you also don’t need a separate bundle for this. Conclusion - a ready-made bundle is not needed, since everyone has their own requirements.

I do not understand why phpbb3 did not fit, especially since the installation seems to be quite simple. phpbb4 will not be released soon, I think.