Holy smokes!!! Your child is so incredibly advanced! I usually describe my son as "different" rather than bright. He's not the kind of kid who answers all of the questions correctly. He's the type to ask questions about the given questions and come up with his own

. I worry about his future in our public school, and am also hesitant to try acceleration.
I already turned down the offer from the school to assess our child for acceleration. Like your child, ours has been described as immature (throughout preschool, which he had a very hard time with), and we have wondered whether he could survive the longer kindergarten day (2 hours longer than pre-k; our picky son has never eaten a meal at school before and gets very tired). The first grade school day is 3 hours longer than his pre-k schedule.
He was in a play-based preschool for 3 years and never understood that school was a place where you were supposed to learn things like reading or math so in some ways he is not feeling frustrated by the curriculum yet. At first, I was concerned because of his radical under-performance. The teacher recommended an online reading and math program that is aligned with the common core standards to give us a better idea of where my child stood. I think she wanted us to learn how exaggerated my own perceptions were. Instead, my son breezed through the content and completed 75% of the second grade reading curriculum and 80% of the second grade math content in less than 2 months with occasional use.
Outside of school, my son has been incredibly driven to learn--learning letters, colors, numbers, shapes, memorizing long stories, reciting the alphabet forwards and backwards, spelling his name, exhibiting a freakishly large vocabulary and advanced expressive language as a toddler and teaching himself to read despite our concerted efforts to steer him into other interests. I raised concerns that he will be reading and doing math at a third grade level at minimum when he starts kindergarten. This is when we were offered acceleration by our school principal...
I was very compelled by the prospect of the networking and extracurricular enrichment opportunities offered by the Davidson Young Scholars Program. I am also very interested in the advice a family consultant could provide us. Most of what I've read about acceleration seems to suggest that it is bad for kids, especially for those who are having a lot of problems connecting with peers socially. The matter is even more confusing given my child's probable ADHD diagnosis.
Our main reason for achievement testing--now it may sound silly, but it would be to support our application for the Davidson Young Scholars Program given that his IQ test took place 7 weeks before he turned 4. The achievement testing costs about $600. I'm not sure if my child would test really low since he is self-taught and has many holes in his learning (like grammar, which he has asked questions about because of his online reading program, but has otherwise not been "taught"). He was oppositional and argumentative during his WPPSI, and so I worry that we might pay for testing and put my son through an ordeal only to have him under-perform or fight the test.
I am primarily unsure about which test would be best given his age and possible ADHD. Are some of these tests normed more for the younger end of kids? Will any of these tests be harder for kids with ADHD?
The achievement tests I am asking about are the ones recognized by the DYS program. We are not interested in acceleration at this time. My gut tells me to let my son play for another year and then maybe learn about the school routine in kindergarten. I am planning to put him into a dual English/Spanish immersion program so that he can experience learning the alphabet and reading with peers, although I have no strong conviction that the dual immersion program will be able to address his asynchronies, especially if he starts reading at a very advanced level in Spanish too.
Thank you for your insights. I'm so impressed with how some of these kids are navigating school. It is actually blowing my mind. Our preschool teacher for 2 years was a very strong proponent of staying with your school/age cohort. I trusted her. Preschool was a nightmare in many ways with a few bright spots. I wonder if I could have changed things by going with my gut and putting him with older kids before his spirit got crushed by his school-aged peers who rejected and teased him (when they finally got old enough to start talking--it seemed to take two years before his peers were talking enough for him to start talking to them, and then things got very sad in whole new ways).