Badaling Hotel. The private tour including the sightseeing trip to the Great Wall and the hotel rate? Usually, it is not necessary to stay one night there. Tour ton Great Wall usually include the transportation and admission fee between the city center and Badaing. Here are some Beijing tours:
https://www.travelchinaguide.com/package/beijing.htm
If you mean other part of the Great Wall, I suggest you stay in the downtown area of Beijing and have day visit to the Great Wall.