We did it last year in last week of June, so right in the middle of what you are asking about...sorry. 5 Kids 11 and under and 2 teenagers. It was the first multiday for the oldest. The bugs are real, as others have mentioned, as well as the hot temps.
We froze some orange juice containers of water and buried them at the bottom of the cooler. It is amazing what a drink of lemonade with ice on day 5 can do for a kids attitude
We were right on the backside of high water and some of the smaller "rapids" as designated on the river map were completely gone, meaning you had to pay extra attention to where you were. The ducky with the two teenagers swam Joe Hutch, long story that no one was happy about, but everyone was OK. The group behind us flipped a boat in there, so don't blow it off. For that reason alone I'd go later in the year with kids and kayaks so that you get some lower water.
It will make for more/bigger campsites as well. Also get a tribal permit for camping river left. The camps aren't as clean (read as cow patties) but the cost is worth it just so you have options.
And final bit of advice, we were rocking an umbrella for shade and I ended up fighting that &%$$ thing too much. I'll be installing a bimini before I do that again.