(most) gamers are particular about latency. but you probably also want a wireless headset for convenience. both my kids use XM4s for comms, gaming, and music. later sony models (and a bunch of others) would also let you do all those - wired via 3.5mm.
newer (more expensive) models support aptX LL, but you might need a bluetooth dongle that supports that, too. some also let you use it wired via USB connection. current B&W models (aptX + USB), for example. expensive, but musicality shines. Note, the Px7 S3 gives you better bang-for-the-buck than the Px8 S2!
at the price range you mentioned, go and test what Edifier offers. the Wxxx or ESxxx series would serve, but they also have a gaming lineup - Gxx series.