[officialtony]

I don't believe that just because the BR takes off for 1st when the ball goes into the dirt means it is a 3rd stikre. Many teams are taught to run to 1st at that time and let the umpires sort it out. If the catcher throws the ball up the right field line - his mistake.
I think the real concern occurs if it IS ball four/strike three and the runner runs past 1st base. A heads up F3 may tag the runner after he passes 1st. Now if you appeal to your BU immediately, and the BU says " no he didn't go ", you have ball four and runner is out on the tag. If you appeal immediately to your BU and he says " yes he went " and the runner has beat the ball to the bag but got taged by F3 after he overran the bag, he is safe. The immediate appeal allows the BU to make a correct call at 1st when all this happens in a second. If he calls him safe on the tag, then you appeal and he says " no, he didn't go ", he is now out on the tag - and the conversations begin. I agree 100% with Bob J. If you call ball as PU, immediately go to your PU if the ball ends up in the dirt. You're not throwing him under the bus, you are doing the right thing for the sake of the game.
Windy way of saying make the immediate appeal to your partner. Job done right.