I see the Infineon (Cypress)
CYBLE-212006-01 Bluetooth 4.2 module.
2-3.6V and 90mA TX full power, 17mA TX quiescent, 8mA RX quiescent, 256KB FLASH 32KB SRAM. MCU 48MHz. Thing is expensive $10 but includes the BT stack, it's been around since 2017.
They are likely using the platform stock firmware in the module.
I would look at serial debug info on the comm header to troubleshoot. I don't think the DMM LCD has much to offer for that as far as displaying connection error messages.
The 3-pin header J10U is just battery power to the BT board LDO. Two pins +, and one or two pins for GND.