I was searching for headsets looking for a good one and stopped when I reread your post and you said you wanted to go wireless. However, I did come across a couple that I liked.
It's just outside of your budget at around $160 (with tax and shipping, probably around $170 or $180), but not terribly pricey compared to some other wireless headsets. It's the HyperX Cloud Flight. It is a very new headset, so there's not a lot of reviews on it right now. The only real downside I see to this headset is the lack of Surround Sound. To be fair though, my current headset has it and I rarely turn it on, so it's only a slight con as far as I'm concerned. Apart from that, it's a good buy. Long battery life (30 hours with lights off), great comfort, detachable microphone, and I personally like the design. It's very basic, but I think basic looks good.
The other option I saw is within your budget. Corsair Void Pro coming in at $100. It's honestly hard to find a negative review or a con that's not just someone trying to find something wrong with it. It has simulated 7.1 Surround Sound, is said to be very comfortable, and the only con I see to it is that battery life is a little low. 16 hours with LEDs off and only around 6 with them on. However, for the price it's at, it seems like a good choice. Your color options are White or Carbon, so up to you. I personally think the White looks badass, but up to you.
I'll link them both, so you can take a look at them. I'll also keep looking to see if I find something better in your budget.
HyperX Cloud Flight:
https://www.hyperxgaming.com/us/headsets/cloud-flight-wireless-gaming-headset
On Amazon:
https://www.amazon.com/HyperX-HX-HS...517238047&sr=1-1&keywords=HyperX+Cloud+Flight
Corsair Void Pro:
http://www.corsair.com/en-us/void-p...ing-headset-with-dolby-headphone-7-1-white-na
On Amazon:
https://www.amazon.com/CORSAIR-VOID...7236853&sr=1-2&keywords=Corsair+VOID+Pro&th=1
In case you were wondering at all during this post, I have a Logitech G933 headset. Most comfortable headset I've ever worn and I love being able to plug my phone into it. I don't have to stop gaming at all to take calls anymore as I just plug my phone into my headset and can hear everything perfectly. However, the mic on it went out within a year and I had issues with it before anyway. I had money to burn, so when the mic went out, I got a Blue Yeti. I personally love my setup now. But, the price of the G933 is too high to justify the amazing audio quality and the horrible mic.