Why can't I use RFID reader and BLUETOOTH MODULE on SERIAL3 at the same time?

I have OSEPP Mega 2560 which is arduino compitable. I'm just most of you knew that. I am using it to test the project I am working on.

[Project] I am removing factory immobilizer in my car and want to replaced it with my custom module. I am taking care of rolling code, and all the other stuff. Why? I want to have a remote control/start but do it my way.

[Problem] My Bluetooth HC-06 module and MIFAIR RC522 don't want to work together. They both work fine separately, but not together. Bluetooth module's TX/RX pins are connected to mega RX3/TX3 pins. RFID module is connected to the pins described in example sketch file. I also tried multiple different-ish sketches and all of them work as long as hc-06 is not connected.

Here is the library I am using. https://github.com/miguelbalboa/rfid Any hints why both modules don't want to work at the same time? Edit: I also realized that when testing bluetooth and rfid module at the same time, arduino was connected to computer for serial monitoring. Could it be that pins from TX3/RX3 or any other pins used by RFID module are also used for pc connection?

Offtopic: Why I need RFID module. This is plan C in case remote start and other plans fail. When module reads valid rfid tag it enables accessories and ignition lines. Then it allows the starter to spin with push-start button.