Surely it's a bit of backwards logic to ask for a CMS to do what an LMS does and more, because you feel LMSs aren't flexible enough?
They're catered to a task, you're not going to find an out of the box CMS that does what you want if you're unhappy with that - it'd be more practical to just make your own.
Maybe if we knew exactly what you wanted that LMSs weren't providing? I'm unsure of your setup but in my experience the whole point of an LMS was so intranets didn't have to deal with multiple CMSs.