The entrance to Machu Picchu will be included in the cost for the Inca Trail tour (do check that when you're booking the tour). The tour company should be okay with and able to arrange for you taking transport back a day later. (I don't think they'll give you a discount for not taking transport, and arranging transport yourself can be costly.) Accommodation in Aguas Calientes for that night is something you'd probably best arrange yourself (there's a ton of options), although the tour company would no doubt be willing to do that for you and take a cut.
I hope you won't regret the "straight to Cuzco" decision. At the very least give yourself plenty of time there before setting out on the Inca Trail, and take things very easy on your first day in Cuzco.