I kept my
boat at Green Turtle for a while, and yes, it's generally cheaper to fly directly into Treasure Cay since taking a Taxi from Marsh Harbour is fairly expensive. ($70 two years ago when I did a crew switch in G.T. instead of Marsh Harbour)
I booked my flights through AA, but there are other carriers. I'm sure all the popular travel booking sites will provide the options on the days you are looking for. I've almost always flown through
Miami and never
Nassau.