LOL, he likely spent more money having you put in that transformer than it would have cost for a 480V motor, plus now he has extra losses in the system that will increase his electric bill. But oh well, at least you got a contract eh?...
Do not confuse the "Y" of your power system with the mention of "Y" connections on the motor, those two issues are totally unrelated. That motor is designed to take 200V, which is the correct "utilization voltage" for a 208V 3 phase distribution system, which is exactly what you have now. Just run 3 wires to the motor and be done with it.
Whether or not you need to use Y-Delta starting is also a separate issue, having to do with the load causing a voltage drop or not. At 15HP from a 150kVA transformer, I seriously doubt that will be an issue!
As to "Do I need a motor starter for this unit?" question, I am going to HOPE you meant "Do I need a Y-Delta motor starter for this unit?", because you ALWAYS need a motor starter. But no, as I said you do NOT need a Y-Delta starter in all likelihood.