My CO pays all fees and provides every Pack with a somewhat flexible budget for awards, events, adult training, and a limited number of local/district/council over-nighters. Boys pay for their own uniforms and books, and long-term camps like Cub Day Camp and Summer Camp, with help and options available to them if needed. Families are expected to pay as little out-of-pocket as possible, and no boy is turned down from any activity or program if they don't have the means; such cases are covered by local Church leadership. The same benefits are available to all boys in every unit, whether they are members of the Church or not.
While I firmly oppose BSA's girl decision, I strongly support kindness and sensitivity in personal interactions. Even though their Cub Scout Pack is at fault for breaking fundamental rules about mixed-gender Dens, a kind approach is still merited as the situation gets resolved. A Scout is friendly, courteous, and kind, and nobody should be made to feel like an outcast.
First, I'd explain in the friendliest way possible to the girl and her parents that we are excited for her interest in Scouting. Then I'd also explain in the friendliest way possible that because we are a boy-only troop, we are not structured with the right organization and leadership to provide her the Scouting experience that BSA has designed for girls (providing as many or as few supporting details as they like). Finally, I'd offer assistance to help her find a girl-only troop or a linked troop in the area that *is* structured with the right organization and leadership to provide her that great Scouting experience (with an explanation about the rollout beginning in February 2019).
If these good-faith gestures made in friendliness are rejected, it would seem clear to me that this girl and her family are not looking for a solution - they are looking for a fight.
There is far too much unsaid in this episode for any of us to fully understand what transpired here. I will say that it is unfortunate that the SPL, who is a youth leader, had to come between to adults. That should never have had to happen. And there are always two sides to everything; we have parts of your perspective on what happened, but nobody from the opposite point of view to give us a more balanced sense of what really transpired. So we can't really make any calls on the episode itself, except that things got out of hand to the point that a young man had to intervene when two adults lost control of themselves, and that was clearly unfair both to him and to any youth who may have been witness to it.
Now, you ask how a Scoutmaster can be objective about other issues when he cannot be objective about other people. But I think you are asking the wrong question, or at least it isn't a question that will bring you any useful answers. The simple fact is that being uniformly 'objective' about all things is not a requirement for serving as Scoutmaster, and it's unwise and unfair to expect such from anybody. Scouting is run by people, and people have failings - you, me the Scoutmaster - everybody. Right now, your Scoutmaster doesn't feel comfortable with your presence in the Troop. That is the fact of the matter right now. Maybe this is totally unfair, but then, maybe there is a good reason for it. His prejudice against you may be one of his failings; his reasons for it may be some of yours. But you'll get further asking how you can change and improve your own attitudes and behaviors than you will by questioning somebody else's; after all, you have total control over how you deal with this situation, but frankly, you'll discover you are powerless to control how others do. If you approach the Committee demonstrating a willingness to temper your feelings and make whatever changes are necessary to make them feel comfortable with your continued role in the Troop, it will make a far better impression on them than any accusations or attempts to justify your behaviors will. And it will teach your children a far better lesson as well.
One thing I have learned working with children is that it's never productive to tell an angry child to "calm down." Nobody wants to be told how to feel; they want to have their feelings acknowledged and respected. Rather, ask them to explain their feelings while you demonstrate to them what calm is through your own example (catch: this actually requires you to remain calm). Questions like "Why are you feeling upset?," "Can you help me understand the problem?," or "What do you think I can do to help fix this?," asked calmly without demanding calm, can do far more to generate positive results than simply telling them how to feel.