I don't have any solutions to add to the conversation, but maybe can add some perspective:
https://youtu.be/TdewERComl4?t=2m35s
2013 Einstein.
Our alliance was officially eliminated from the FRC Champs after the loss in this match.
It turns out that a very well-meaning volunteer had miscounted the number of discs scored. They wrote down 64 discs scored instead of 46 on the scoresheet for our opponents, and we lost the match.
Although it's not identical to your situation, it has parallels.
We made our case to the refs, multiple times. We were turned away, multiple times. In fact, at one point we fully accepted that our chance at a World Championship would be taken away due to a scoring error. We pulled our robots off to the side, put our tools away, and reflected on the end of our Einstein run. Shortly after being eliminated from Einstein, we took this picture:
This picture still amazes me to this day. The smiles you see there are genuine. We just had one of the worst possible things imaginable happen to an FRC team: we were officially eliminated on Einstein due to a scoring error! Yet we were still happy that we left everything we had on the field, and were so proud of what he had accomplished - even if we didn't get the chance to move on.
Had our champs ended right then and there, I actually think we would've been okay with it. Genuinely. Call us weird, but I think that is 100% a reflection of the type of people and teams we were/are.
Of course, we didn't exactly give up, either. With the robots for Finals 1 already lined up on the field, our students made one final request to the refs. We bolstered our case, not with video as the rules were very clear on this, but with mathematics. We ultimately presented a "proof" showing that the score was mathematically impossible based on the autonomous score, pyramid points, and the number of unscored discs left on the floor and in the human player stations at the end of the match.
The refs congregated.
The scorers congregated.
Something was happening.
They found the scoring error where 64 was written instead of 46, the volunteer responsible confirmed it, they corrected it, changed the outcome of the match, and pulled the robots off the field to play a deciding SF match instead of the finals.
So yes, this is an example of FRC "getting it right" (although I am sure the situation left a bitter taste to the alliance of 3476, 1640 and 303 - and who could blame them), but this was a very difficult situation that I have to give FIRST and the volunteers a lot of credit for.