"They delved too greedily and too deep, and disturbed that from which they fled, Durin's Bane. Of what they brought to light the Orcs have gathered nearly all, and given it in tribute to Sauron, who covets it." (LR II v)
We can assume they gave him the knowledge of the awakened Balrog as well; Sauron very probably therefore.
On the topic of the Balrog being actually sent by Sauron, see HoME VII:
"'No Balrog has lain hid in the Misty Mountains since the fall of Thangorodrim,' said Galadriel. 'If truly one was there, as is told, then it is come from Orodruin, the ountain of Fire, and was sent by the Lord whom we do not name in this land.'
In FR, of course, the view expressed here by Keleborn or Galadriel that the Balrog, sent from Mordor, had entered Moria not long since ('it is come from Orodruin') has no place. In LR the Balrog of Moria came from Thangorodrim at the end of the First Age, and 'had lain hidden at the foundations of the earth since the coming of the Host of the West' (see pp. 142 - 3)."
This idea was thus abandoned early.
It is indeed that the case that Dúrin's Bane would owe Sauron no allegiance, even though they were clearly inferior in spirit -- "Sauron is just another (if greater) agent. Orcs [read: even Orcs] can rebel against him without losing their own irremedable allegiance to evil (Morgoth)." (MT VIII)
The rest seems to have to remain speculation.
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