bluetooth chip is bc417. it draw 30-40mA current when not paired and about 5mA after pairing. it have somekind of sleep mode but i dont know how sleepmode works.
i noticed if voltage for bt module drops below 3v it is not stable anymore so it perhaps dont work well with 3V coincell.
use google for datasheet >> linvor datasheet or hc05 datasheet
JY-MCU use 5v and have 3.3v regulator for module.