Playable Balrog in The Lord of the Rings: Conquest (Xbox 360)?

No. Balrogs are not playable. The roster includes:

• Free Peoples: Aragorn, Legolas, Gimli, Gandalf, Éowyn, Faramir, etc.
• Sauron’s Forces: Sauron’s Lieutenant (Mouth of Sauron), Nazgûl (Witch-king, Khamûl), Orc Captains, Trolls.

You fight Durin’s Bane (the Balrog of Moria) as a boss in the “Mines of Moria” campaign, but cannot control one.