HenrikBuhl, What is the baudrate you are using? You are only acquiring 1 byte from your eeprom. I to use a 18F6720 with 2 24LC1025's, 2 active ASM driven comms @ 38.4k baud and I read 128 bytes or a page when I turn off all interrupts momentarily. The usart has a 2 byte buffer so if your baudrate is relatively low you should not loose any characters in your comm's. By the way you mention that your LCD dimming flickers. I to have an LCD on my product but I use the CCP1 & CCP2 PWM module's for the contrast as well as the backlight dimming. That way there is a continuous pwm signal to the circuitry + you have 1024 steps of backlighting.

Dave Purola,
N8NTA