[Eldy]

If Durin's Bane was under Sauron's control in the Second Age, then I'd think that Sauron would have ordered it to do something other than hibernate for 5000 years. In all seriousness, even if you don't think the Balrog was literally asleep for all that time, he was stuck underneath the most powerful Dwarf realm in northwest Middle-earth, one that Sauron was unable to force his way into during the War of the Elves and Sauron. There was no opportunity for them to be in contact since the Balrog had already gone to ground by the time Sauron started dipping his toes back into the Dark Lord game.

[Edit: I went back and double-checked Appendix A and it's pretty clear that the Balrog was asleep for most of its 5000 year disappearance, though it's left uncertain whether the Dwarves in T.A. 1980 were the ones who woke it, or if the Balrog had already been "awakened by the malice of Sauron" but remained trapped in "prison" beneath Khazad-dûm. Either way, it was out of commission during the much earlier time period that the "gathered under his government" quote refers to.]

I agree with Huinesoron's points about the evident limits of Sauron's control over evil creatures such as Smaug in the late Third Age. Note also that the Moria orcs served neither Sauron nor Saruman (cf. the chapter "The Uruk-hai"), which would be a weird setup if they were sharing the mines with a Balrog in the loyal service of Sauron.