UART no reply from BTM511 Bluetooth module
The Bluetooth module seems to be in some sort of low-power mode, where it powers down the UART circuit when its RX pin is LOW. Because the LA doesn't keep the RX of the module HIGH, the module would shut down its UART.

It takes a short while after the RX is HIGH for the UART to start (about 0.7 seconds from my measurements), shown by TX getting HIGH. So after setting the RX to HIGH, you have to wait before sending any commands. (couldn't find this anywhere in the datasheet though..).