caught or trapped?

Question
This happened last year during our rec league tournament. We won so we dropped the protest. We use NFHS rules where ours don't cover something.

Bases loaded, two out. There was a line drive to F4. She jumped to the side, caught the ball with some yellow showing at the top (ice cream cone), fell to the ground with her glove in the air, rolled onto her back and jumped up with the ball still showing through the top of her glove. The BU, standing less than 10' away, called the batter out and our team ran off the field. Unbeknownst to anyone other than the offensive 3rd base coach, the PU said she trapped it and proceeded to allow the offense to all run the bases and score.

My argument was the BU was in position and it was his call. The PU insisted he could over rule the BU and it was his decision that she trapped the ball. To that my point then was, "in that case the play was dead. If the batter was safe then all runners could only advance one base as the defense had relied on the BU saying the inning was over." Even the offense had started running off the field until the PU told the coach it was a trapped ball. He would have none of that and insisted it was the PU's call and his was the one that mattered.

We still went on to win the game so I never followed up by filing the protest. I believe it was the Base Umpire's call, he may not be over ruled, and when there is a disagreement between two umpires the play is dead.

Any comments?

Answer
Hi Pat,
Thank you for your question.  I like your honesty.  You came right out and confessed that you had a question about a play that happened last year.  This play has been nagging you for more than five months, at least, so why don't we put your concerns about it to rest?  

Your question actually isn't about whether the ball was caught or not (trapped), it's about the dynamics between the plate umpire (PU) and the base umpire (BU).

The play became a "clusterintercourse" when the BU ruled on whether the batted ball was caught or not.  Obviously, the BU didn't "go out" or "chase" on this ball hit to the INFIELD, so the decision whether the ball was caught or not RESTS ENTIRELY with the PU; this is the PU's call!  It doesn't necessarily matter that the BU was fewer than 10' from F4.  The PU still may have had a better view and/or angle.  The BU should have just observed the action and (silently) judged whether the ball was caught or not.  

I'm 100% certain that the defense ran off the field due to the inappropriate "out" call by the BU, and because the PU's "no catch" call was so weak that reportedly only one of the game's participants noticed it, the 3B coach.  So now the umpires have a big mess to clean up.  Ideally, the umpires should have put their heads together.  The PU should have asked the BU whether the ball was caught or not.  If the BU makes a good case, then the PU reverses his call and the result of the batted ball is a catch and a simple third out.  If the BU doesn't make a good case, then the PU stands by his call, the result of the batted ball is a no catch, the batter/runner is awarded 1B, all runners advance one base, one run scores, and play continues with two outs.  Not ideally, and that is a gargantuan understatement, your result occurs and the PU sticks to his guns and allows all four runs to score, and he may allow that, since the ball never did become dead.  

In summary, addressing the last sentence in your question, it wasn't the BU's call, the BU may be overruled, and the ball doesn't become dead (in this scenario) when there is a disagreement between two umpires.

Scott Kelly