My opinion (and experiences).....
It'll be hot and crowded during your trip. I'd go to the MK each morning and do any and all of the rides that your kids can and want to do.
Spend the afternoon napping and relaxing, and then do the Studios and or Epcot at night.
AK is a wonderful park, but you can't escape the heat in the summer. I personally wouldn't spend the time trying to see AK in the heat with the little ones.
My kids have always gotten the most fun out of the MK as toddlers. Now that they are 7 and 5 (the older ones), they have more fun at MGM (can ride ToT and Star Tours, and they really like Muppets now). We've done AK, but never in the heat. For example, we just returned from 9 days at the Polynesian--we went to MK, Studios, Epcot, Disneyquest, pool, etc., but didn't go to AK once. We WILL go to AK this December on our return trip.
Remember too, that travelling to the parks with kids is relatively slow--I'm assuming you'll have a stroller and diaper bag--the bag will be inspected at the entry to each park. Don't think of walking from the hotel to MGM in June/July--it can be done, but the heat will slow you down and, probably tire you out for the rest of the day.
BTW, the WDW website says that on June 30, the MK will be open til 11 pm and there will be 2 SpectroMagic parades that night. You can count on huge crowds at the MK that day, so make sure to arrive early.
Good luck!