A CR2032 Lithium battery is designed for applications that have about 200uA current draw. Even at about 3ma, their voltage drops to about 2.6V. The Arduino, even in low power mode will draw close to 20 or 30 ma while the BT module it not active and nearly 200ma when transmitting or receiving. That is way, way too much for the CR2032. Try a couple of AA cells.