tl;dr
I can't send data using the raw SPP-C Bluetooth module without a breakout board. The communication via UART works fine as I am able to issue AT commands and get proper responses. But when connected over Bluetooth the data passed over UART does not show up on my computer while the other direction works fine. Has anyone tried that module yet without using a breakout board?
The long story:
I started my project by using a regular HC-05 module on a breakout board. The prototype worked just fine as there are also plenty tutorials available. The drawback using that breakout module is that it works with two voltages, 5V for power and 3V3 for communication. I have also more components in my project that run best at 3V3. That is why it came to my mind to use 3V3 as the only voltage. The raw HC-05 module without breakout board is also powered with 3V3. So everything looked just perfect.
Looking at the specs to the HC-05 I noticed that there is also more on that module like SPI and USB (page 12 bottom). So I got curious and off track.
Anyway. Instead of ordering raw HC-05 modules without breakout board I ordered cheaper SPP-C modules that don't have all this fancy stuff but just UART communication.
Following the description from Martyn Currey I was quicky able to issue AT commands and receive data over bluetooth. But I am not able to send data over Bluetooth.
I am using the same setup as it is shown on page 4 in the Chinese datasheet. (no I can't read that crazy symbols but I can read and understand the images)