Hey there, sounds like a lot, especially in the heat of August. But Day 2 'just' seeing the Vatican is very wise. So I'm going to say the opposite of some others - NO WAY would my family tolerate visiting the inside of the Colosseum or the Forum on a hot August day. I"d suggest checking out the Capitoline Museum first. From the square between the 2 museum buildings, if you walk straight ahead and slightly downhill, there's a great terrace that looks out over the Roman Forum. OR go to the cafe on the top floor (you can access without tickets, but much easier to take the lift.) From there the views are magnificent.
What about a sunset golf cart tour or walking tour of the historic center the day you arrive, when everything is lit up and it's slightly cooler? The area between the Colosseum and the Capitoline museum is lovely at night, when it will be cooler.
And if you can get onto the Pantheon website that day after you arrive, you could possibly book tickets when you know what time you'd be there? OR wing it and book those tickets ahead, as I don't think they're very expensive. Walking from the Capitoline to the Pantheon isn't far, maybe 10-15 minutes. The Pantheon is cool inside, not a long visit, and waiting in line without tickets could take 45min or longer, in the HOT sun. https://direzionemuseiroma.cultura.gov.it/pantheon/.
Where's your hotel and what's for dinner?? LOL, getting to sit down and say ahhhhh over a good meal would ease the stress of the day. Good luck!