Did Galadriel & Nenya keep the Balrog from Moria out of Lothlorien?

[tifo_gcs]

I don't see why the Balrog shouldn't have wanted to venture forth at some point. Especially after Sauron's return to Barad-Dur, and the tales of evil things stirring once again, his gathering of the forces of darkness, he must have known where the Balrog was, especially with the orcs also in Moria. Then one can only suppose that he left it to guard Moria. I don't believe that the Balrog would refuse the order of Sauron, Sauron having been Morgoth's right hand.
Given the time it took for Gandalf to finish off the Balrog, I think that had the Balrog tried to attack Lorien, which in plain military sense is stupid due to lack of any form of support, then a struggle between the Balrog and Galadriel ought to have left obvious traces in Lorien.
So I don't believe that the Balrog ever ventured outside of Moria after it settled down there, and why is anybody's guess