A CR2032 can be inserted into the battery socket of the Wireless starter kit, but the maximum current of CR2032 is about 10mA. This current will cover the current consumption of the BGM1xx transceiver, MCU core, peripherals, and external circuits.
For example, in the case of BGM113, the consumption current value of the transceiver part is 8.7mA during reception, 8.2mA@0dBm and 16.5mA@3dBm during transmission. MCU core running current @EM0 is 88uA/MHz * 38.4MHz=3.3mA.
It is not very realistic to operate with only a coin battery.

In addition, the user guide of each radio board also has the following description.
Note: The current sourcing capabilities of a coin cell battery might be too low to supply certain wireless applications.