Personally I don't see why not. Looking down from the balcony isn't gonna hurt anyone, and the ops room itself is so darned BIG a few visitors at a time probably wouldn't hurt either.
I have memories of crowds of visitors around the old Worthing Suite at LATCC (sniff!) which definitely were not always welcome, particularly when it was busy (most of the time). The same cramped conditions don't apply to NERC.
However, while the operational ATCOs and ATSAs probably won't have a problem with a few aviation type visitors, the management will take a different view.
You see, you may actually end up talking to the staff and finding out the truth about NERC and that would not do, would it? You might get to hear first hand how bad the RT is, how poor the Headsets are, and just how difficult it is to read what is in the FDE / FFP windows. Then the truth would really be out and I'm not sure management are prepared to be so open just yet. At the moment they can pass all the problems off as "the moans of just a few" (see the NERC..so far thread).
Anyway here's a plan for you. Ring the LACC switchboard (01489-572288) and ask to speak to someone about visiting. Explain who you are and why you'd like to come. When they say "no way", just say thanks and hang up. Then ring the number above again and ask to speak to the ATC Watch Managers DESK. Ask again and you might notice a complete change of attitude.
No guaratees, but give it a go and let us all know how you get on.
Rgds BEX