I got a nice pair of Turtle Beach wireless for under 150
Turtle Beach Stealth 600 Gen 2 - Auriculares Gaming Inalámbricos - Xbox One y Xbox Series X, Negro https://www.amazon.es/dp/B08D44W56B/ref=cm_sw_r_cp_api_glt_i_4781TT2M3Z9ZTFTN8G1H?_encoding=UTF8&psc=1
Same as the controllers there’s a pair button and it hooks up no problem. It’s made specifically for Xbox One and even works with PC. Here’s a link: https://www.amazon.com/Turtle-Beach-Stealth-Wireless-Gaming-Headset/dp/B08D44W56B/ref=mp_s_a_1_4?dchild=1&keywords=xbox+one+wireless+headset+turtle+beach&qid=1615330053&sprefix=xbox+one+wireless+headset+turt&sr=8-4
Didn't know this was out: https://www.amazon.com/Turtle-Stealth-Wireless-Gaming-Headset-Console/dp/B08D44W56B/
Looks good for your budget. You'll notice people complaining about it having issues in the reviews. While others say they had no issues after updating the headset before using it. So keep that in mind.
The new Turtle Beach wireless Stealth 600 gen 2 model comes out on September 20th (at least in the UK), and looks pretty well specced for the price (£89.99 / approx 110USD?).
https://www.amazon.co.uk/Turtle-Beach-Stealth-Wireless-Headset/dp/B08D44W56B/
I use wireless headphones that connect directly to the Xbox: https://www.amazon.com/gp/product/B08D44W56B/ So, when I use the TV audio, it comes from the HDMI cable. If I use headphones, it is wireless. The optical cable only drives the bass shakers. I don't know if there's a work around to your set up...
I recently picked up this turtle beach headset for the series X and have been impressed: