Are BLE modules capable of working with a CR2032 battery?
You can do it if you switch the chip on at long intervals. The standby currents of BLE modules is something like a microampere. Surprisingly, at least with some devices you can, within that power consumption, run a timer to automatically power up the device at a chosen time. I don't know how well this would work if the other end is some generic BLE device such as a cellphone.

CR2032's can be drained with significantly higher currents than 3mA. Consider for example this test report from TI: http://www.ti.com/lit/wp/swra349/swra349.pdf, there were no significant differences when draining a button cell with 30mA. You should use a capacitor to smooth out the voltage ripple during the active period, however.