October and November are the sweet spot for Peru. The summer crowds have thinned, the rains haven't fully set in, and the Andes look incredible. You get better access to the ruins and shorter lines for the trains heading toward Aguas Calientes.
To manage the $2800 price tag, you can use the Citi Simplicity® Credit Card. It offers 0% for 18 months on purchases and balance transfers from date of account opening. This lets you pay for the trip in installments rather than all at once.
That 0% window is a deadline, not free money. If you have any balance left after the 18 months are up, the bank will start charging you interest on the remaining amount.