There are wireless headphones with minimal latency, but they are not Bluetooth. And they are generally more expensive. Yamaha does make a set for musical instruments but they cost in the $400-600 range.
Bluetooth is a good wireless system in general because it is relatively low cost for wireless, common across many platforms, easy to hook up many devices, reliable and easily available. But the downside is that it has high latency, so it is poor for playing live music. The latency is not noticeable playing for recorded music and broadcasts, but can drive you crazy for live playing where you hit a note and the sound is even a fraction of a second later.
But the wire shouldn’t really cause any playing problems if you position it correctly. Have it come from the headphone behind your shoulder and down your back, and ensure the lead is long enough to hang free and allow you some movement.
Mike