Those LEDs are controlled by the USB Core code, and are defined in the board variant. For the pro micro that's the pins_arduino.h
file in the micro
variant:
#define TXLED0 PORTD &= ~(1<<5)
#define TXLED1 PORTD |= (1<<5)
#define RXLED0 PORTB &= ~(1<<0)
#define RXLED1 PORTB |= (1<<0)
#define TX_RX_LED_INIT DDRD |= (1<<5), DDRB |= (1<<0), TXLED0, RXLED0
If you edit that file and remove everything after the macro name, so you have 5 empty macros, it should disable the LEDs completely.
#define TXLED0
#define TXLED1
#define RXLED0
#define RXLED1
#define TX_RX_LED_INIT
Edit:
For the Sparkfun Pro Micro, edit the file at
...\AppData\Local\Arduino15\packages\SparkFun\hardware\avr\1.1.13\variants\promicro
This is because it has an extra package that is not installed by default.