Wired will be better period. Several Choices for 400. I'm a fan of Sennheiser. I have the 650HD cans. I like them a lot. I also have the 800s, but those are way expensive and I can tell you for the cost the 650s are 90% of the 800S cans. Invest in a 15ft grado extension cord and you'll have plenty of room to move around if you are using these in the living room for music and movies. My headphones are open back which is my preference. Others may chime in for closed back options. You might head over to headfi.org and sift through that site. Many reviews posted there of all makes of headphones and headphone amps. Headphone discussions on AVS is an after thought.
Supercharged Song Towers, Oppo BDP-203, Anthem MRX-520, LG B6 OLED
Headphone Rig: Oppo BDP-95, Burson Audio Soloist MKII Headphone amp/ Sennheiser HD800S, Sennheiser HD650