@terxw - sorry, the manual is wrong and unclear here - when you talk to the bluetooth module you should not connect to the programming pins. they are only for programming atmega chip. Must be a copy/paste error when i wrote the manual i think. Looking at the schematics RX goes only from that programming pin to the atmega and not bt module. TX has a input protection diode as well, so you need to swap that one to.
As you see in the image, RX and TX from that programming point is only connected to atmega chip. Try connecting them to ordinary input. In worst case scenario, the diodes are making the serial communication fail at the baud rate you are testing... (should work with 115200 since thats what i set it to) but worst case scenario, solder wires to BT module and test.
Also dont power it using BAT (it goes through a dc/dc booster if you do). Try a regulated 3.3v power source and input on 3.3v pin.
Also, change the both No line ending to Both NL and CR
Edit: I will be home on thursday, traveling at work, but I have a identical module i build the same time as yours I can test on, if you dont want to solder around and mess the module up.