Welcome! You are fortunate that your district has a gifted full-day kindergarten.
Have you looked into what the gifted program consists of and whether it will be a good fit for your DS? Do you know what the "qualifying" criteria are and whether your DS will need more testing? Our schools have quite the list of tests (achievement and ability) that they give children for G&T ID. Some schools have this information available on their website.
Did your DS just turn 5? I have a DS5, who has so far only been tested on the WJ Ach., because he needed it for a summer program. His scores were terrific (no broad scores were calculated, because they weren't needed at the time, but I have since realized that if calculated, they probably would be DYS level), but I take them with a grain of salt because of his age. Still, he sounds a bit like your son. Just to warn you, my DS5 is currently in kindergarten and is completely unchallenged. I do not think that your average kindergarten is academically challenging for many children. I believe that part of the problem is that in a non-gifted kindergarten classroom, the children come in with quite a range of abilities.
Our schools have no formal G&T ID until 3rd grade. DD8 was ID'd this year and is now in the program. Before 3rd grade, the school has informal "enrichment." DS has received a couple of things from the enrichment teacher, but not enough to really challenge him - the teacher is simply too busy (seems spread quite thin, with other duties) to help him very much, it seems. Luckily, kindergarten is NOT a full-day program, so I supplement reading and math at home (which allows me to work at his pace and cater to his interests). This will not work next year, though, so we will have to see what next year brings.

I do not think it is uncommon for a bright 5-year-old child to have handwriting that doesn't keep pace with their thoughts. Fine motor control is still developing. DS is like this, too. Actually, his handwriting is probably the one thing that he practices at school that I HAVE seen improve.
DS is in K with some neighborhood friends he has known since he was a baby - so he LOVES the social aspect of school. Since he can be shy, already having so many friends at school is great for him. He would HATE to be separated from his long-time friends.
I do not know if any of this helps, but I just wanted you to know that you are not alone. Others here have experience with grade-skips and homeschooling.
Wishing you and your DS all the best!