My garmin watch w/ built in heart rate is always hooked up to my phone via Bluetooth and I can listen to music at the same time.
How old is your phone? Most modern phones can connect to multiple bluetooth devices at once. If yours can’t hook to multiple devices, your cheapest option might be to get a stand-alone HR chest strap + watch to mount on the handle bar (amazon has for around $20) and just use the phone for music