They can broadcast from the central lobby but tend not to anchor from there, I suspect only certain reporters have access.
As for the man shouting, if they move he will only follow, and if they move to a vantage point where he cannot be heard, they will be accused of trying to censor him.
The News Channel and World simulcasted the 6 again as yesterday.
I presume they would have to be a lobby accredited and I'd imagine that would Only extend to Westminster based correspondents and perhaps not all so presenters and news channel producers etc. who are essential to getting the content on air wouldn't be covered.