I think cutting Chengdu make sense. You can do Tianmen Square Forbidden City, maybe visit Jingshan and Houhai and then Temple of Heaven in one day. Save Summer Palace and Mutianyu for another day.
Having a dedicated driver and tour guide will be efficient but there is a lot of walking in these tourist sites, hence we thought your original plan was very ambitious (still do-able), especially for the kids.
I recommend you take taxi to/from airport, price is cheap and subway isn't very luggage friendly, especially at certain transfer stations.