You want to be looking for IEMs (In Ear Monitors) in your searches to yield the best results at that price range.
At $360 they’re pretty far out of your budget but the Moondrop Dusk just got released and they come with a good USB-C DAC cable. I mention them because they also sell the cable separately and will work with most 2pin (non-recessed) IEMs.
I received mine a couple of weeks ago and I’m thoroughly impressed with them and would even consider them good enough to mix with. They hold up well against Senn HD650s which I normally use to mix when I’m not using my monitors.
But if you need to stick to that price range, the Moondrop Aria are also very good for the money (I own them too) and you could get the separate USB-C headphone cable all close to your budget.
I know I seem like a shill for Moondrop but they are pretty much the best price to performance IEM manufacturer at the moment.