I have solved my dilemma. I will spend the night prior to Machu Picchu in Ollantaytambo. I will leave suitcases at the hotel when checking out that morning. Visit MP and take the train back to Ollantaytambo instead of to Cuzco, pick up luggage and take one of the plentiful taxis to Cuzco.

I miss out on the latter half of the train ride but save more than enough to cover the cost of the taxi.