I thought Byrne got booked for diving, not for clipping the defender.
There may have been a foul after that, that could have been a second yellow. I remember thinking "oh bugger" at the time. Can't say for certain it was Byrne though.
Sorry, yes you're right, he was booked for the dive. Therefore, I suppose he could possibly have got a second for the foul, which Carrol seemed to be wanting.