O.K. here's the thing about Punta Cana during that time, it's during their rainy/hurricane season. You're taking a chance late summer. I think that's the wet season for Cancun also. We found that out going to St. Lucia in late October. We happened to hit a good week but the week before we were there and the week after they said was very rainy. They said there were days where it rained all day.
Not a big fan of either Cancun or Punta Cana. If you want to go to the Caribbean during the late summer I suggest Aruba. There's a reason a lot of people go there during the summer. For one it doesn't rain there much so it's always sunny. And it always has a nice breeze from the trade winds so it's not too hot. I was asking someone who lives there and they said the temperature isn't much different between the winter and summer. During the winter the high temperature is 82-83 everyday. During the summer the high is 87 everyday.
Also, there are a lot of great restaurants in Aruba if you're a foodie. Best food I've had in the Caribbean.
The other thing that was nice about visiting Aruba is there really isn't much poverty. Mostly middle class because they make so much money on tourism (80% of their economy). So it's very safe to be out and about there. Nobody ever tried to sell me anything. I went to Jamaica once and I was hounded all vacation. So much so that it was unpleasant.
JetBlue has direct flights there all summer from Boston and JFK. So easy to get to. All the hotels are only about 15-20 minutes from the airport.
The beaches and resorts are great. The only negative is there is no forest or rainforest there because it's so dry. The Atlantic side of the island (east side) is a desert. All the resorts are on the Caribbean side (west side) because the ocean is very calm there.
Let me know if you're interested and I can suggest lots of restaurants and a few resorts to stay at.