My son is moderately gifted with ADHD. He is 9 years old. When he has too-easy work, he has every behavior you listed, plus sitting and staring at an assignment and refusing to do it for hours on end.
I would suggest pushing academics, but at the same time, being prepared to provide accomodations. If he has dysgraphia, writing is exhausting and should be addressed completely separately from academics.
Finding the appropriate level of difficulty is always challenging, as the target is always changing. I would start presenting material without much worry about being developmentally appropriate and see what sticks.
I suggest think of tutoring not as fixing up deficits - like spelling and handwriting - but lighting a fire in him. You are likely his only opportunity for enrichment and acceleration. Don't squander the opportunity fussing about handwriting.
I suggest skipping to 6th grade math. 4th and 5th are very similar, but 6th grade introduces new topics like ratio and proportion. 7th and 8th grade are a slow progression into pre-algebra. Common Core math is an excruciatingly slow progression for a higher IQ child. Don't worry too much about gaps in learning (as long as he's still in public school). My son skipped 2nd, 3rd, most of 4th, he did 5th in 2 months, refused to participate in 6th and skipped straight to Algebra 1. He hasn't had any difficulties. His IQ is not particularly high, barely above Mensa cut-off and math is his weakest subject. I recommend IXL.com for math because you get access to all grade levels, 4K - 12 with one enrollment. This is great for figuring out where a student should be working. It's not unusual for a student to work in multiple grade levels at a time. Maybe a student is strong in geometry and does that at grade 10 but weak in calculations and does that at grade 5. That's normal and OK! IXL does not have an instruction component, it is only practice problems. Khan Academy is a fabulous resource for instruction (and it's free).
I'd point you to Khan Academy for science and history too.
You don't need to worry about a complete and balanced curriculum since he is getting his school needs met at public school. Pre-teaching can backfire hugely later, so hitting the topics schools don't cover well may benefit him much more. Computer programming? Keyboarding! Music. History. Talk to his parents because "afterschooling" can result in acceleration. That's what happened to my son! I afterschooled him in 4K-1st because he was bored and he accelerated so far he couldn't be integrated into a classroom anymore. (Oops). He is radically accelerated and homeschooled now. If his parents don't like that potential scenario, stick to electives.
