From your description I wasn't quite clear what your goals were for your child. E.g. get a higher level math or specific subject or what.
I have 4 very bright children all at different spots on the gifted chart. We have one who skipped multiple grades and others who are accelerated 1-3 years in a specific subject(s).
Initially with our oldest we pursued grade skip, which created a lot of push back in multiple school districts. It seems to have worked out for my oldest but there are trade offs. He will be a 14yo going to GA Tech for BME / pre med starting in May and enter with Junior status. Socially he has sacrificed and will sacrifice a lot. To balance things he plays travel sports at age level.
We have deferred or declined grade skip for 2 of my other ones when we were asked to skip them and instead found schools that were willing to push them ahead where they need it but not by full grade. This keeps them with their peers for part of the day and once they hit high school they can decide to graduate early or take dual enrollment courses at a local college during the last 2 years. These 2 are also potential recruited athletes and will need each and every year till they are 17 to get the size / skill development. If I must skip at this point I would recommend only 1 grade in totality and use course level acceleration and dual enrollment as a supplement. This is the best balance.
A few things you might consider. Just my 2 cents.
1. Seeking out a gifted school where they are used to course level and grade acceleration. Note: We moved schools (public/private) every 2-3 years because fighting for your kids is exhausting on administration and eventually you hit walls that can't be moved. We found being new to a school we could get a lot more of what we want. I had 4 kids and was (am) always battling for more customization for my kids.
2. Supplement outside school. Your child is so young and there is no huge rush. Why not supplement in the subjects you need to outside school. This preserves the momentum while you figure things out. I did this in math with a ton of resources (IXL, Khan Academy, Russian School of Math, tutors, clubs and academic teams, specialty camps and events in your community, buying above grade level text books, etc.). Don't leave all your fate to their brick and mortar school.
3. If your child skipped already there is no rush to push for a double. My oldest I mentioned didn't skip for the first time till 6th grade. He was ready in K but we could not get it done because of multi-school resistance. I didn't stop supplementing and when we found a school willing to help us it all took off like a cannon and he finished 7th-12th (6 grades) in 4 years + gained 2 years of college credit. I wish we could have taken a skip early elementary and then maybe late elementary. 2 grades at once is very aggressive. My son didn't have much of a high school experience really and some days I regret that.
4. In the heat of the moment you are clearly passionate for your child's acceleration but perhaps the school has made some good points. If you make a grade skip you want to be a star not average. I tend to agree on their point of being 75%. I have heard many schools position it this way.
Your child will already have disadvantages in size for sports and socially. You don't want him/her to also be average academically because you pushed a double acceleration.
5. What if your child went to 2nd grade for a class subject (where the data supports he/she should) but remained with the 1st graders for the subjects he is just ok at relative to 2nd graders. Wouldn't that work out well for your child and you. You can utilize summers and supplemental efforts to bridge the gap in those subjects he/she is weaker in to effectively make the final grade jump in a few years.
6. Ultimately if you are the parents and if you feel your child must jump than you could manipulate the system. Pull him/her out - homeschool and return your child the following year with a transcript showing completion. They will still probably test and may or may not accept the upper grade if they don't test for it. If you can find a school that is willing to do the grade skip (even if it is not a forever home) you can go for a semester and then transfer back with a live transcript from an accredited school showing 2nd grade and they should have no choice but to accept it.
7. You may want to ask your child what he/she wants. They are the one living it. Where do they feel more comfortable - with the K or 1st graders and why.
8. You may also want to observe how your child does in the 1st grade and K classes. Once you act as an observer you may see things differently than you are today.
Good luck.
