Just our experience: Both of our girls started K shortly before their 5th bds due to fall bds. Kindergarten was absolutely fine for both of them even though dd12 was reading quite well and really knew everything she was expected to know by the end of the year at the start. Dd10 appears to be HG/2e and she, too, did fine in kindergarten with no special accommodations.
We had a lot harder of a time once they hit 1st grade. That seemed to be the year here where the schools decided to buckle down and drill everyone in phonics, sight words, flash cards, etc. There was no more singing songs about the letter people and coloring the letter of the week. We are in a high performing school district and a lot of kids are reading easy readers in K, so it isn't like K is all play time, but the way academics were presented was in a playful manner which dds didn't mind even if they already knew the academic stuff.
The fun seemed to disapate come 1st grade. Perhaps you can try to get a sweet, fun fit for a kindergarten teacher and then spend K working with the school on a game plan for 1st grade such as subject acceleration.
