I get no response form a known good BT module connected to the RX and TX pins. The BT module and the ATMega328p and are running at 3v. Have you any experience of this configurattion?
You shouldn’t have any issues when running at 3.3V with the Virtuabotix BT2S Slave or Master, I know the VOH at 5V can be a problem on the ATMEGA328P-PU though. I suggest connecting the RX to the TX pin and making sure you are getting communication loopback.
If that works then maybe you can post a picture of your setup so we can debug it further.