The FastLED and NeoPixel libraries turn off interrupts for the specific timing of the WS2812. The Adafruit NeoPixel has a few extra options to be able to still handle some interrupts. The FastLED has a wiki that describes the problem with interrupts: FastLED wiki interrupts
The SoftwareSerial requires interrupts. When the SoftwareSerial is busy, the Arduino is occupied.
To me it seems impossible to make both libraries work at the same time. I think you have to change something. For example use a Arduino Micro or Pro Micro and use the spare hardware serial port. Or use a ledstrip with leds that have a clock signal and don't need specific timing.
If that is not possible, you might have to use pin 0 (RX) and pin 1 (TX), which are also used by the serial monitor and to upload a sketch.
It is possible to split RX and TX. The only thing that RX and TX have in common is the baudrate, but you can use them for seperate tasks.
For example, use TX to send (debug) messages to the serial monitor, and use the RX to receive commands from another Arduino board. Disconnect that other Arduino board to upload a sketch.
[ADDED] After reading the answer by @Andrew and the FastLED wiki about interrupts, I think that my answer and @Andrew's answer are equally good answers. Perhaps the best way is to do it all: faster Arduino board, hardware serial port for commands, do one thing at a time, implement some kind of XON/XOFF in software, and if possible use other leds.