Portable Trombone Warmer Prototype Development

I need a working proof-of-concept for a compact device that pushes temperature-controlled air through a trombone so the horn reaches and stays at its ideal playing warmth. The first build must run from... (Budget: $250 - $750 USD, Jobs: 3D Printing, Arduino, Electronics, Embedded Systems, Firmware Development, Microcontroller, PCB Layout, Prototyping)