You will miss data that arrives while you are servicing the LCD.
since the TX will jump to a label "value" to get ADC values,calculating average from 16 readings, could this compensate the lcd servicing you mentioned?

Another possibility is that the RF signal strength is marginal. leading to sporadic reception.
Maybe i'm wrong here, but i'm starting to think of the possibility that having BOTH RX and TX modules on the same experiment board and having both antenas side by side is NOT the best idea.

Could this be an issue having them so close to each other ?