Since you want to ’test’ the waters, and given your opening post: go over ear, closed back, Bluetooth 5.3 (5.2 at least). BTW, the Bluetooth version needs to be supported both on your phone (for instance) and your headphone (to get the full benefit of the version).
If the pair you go with works to your satisfaction with BT, it will perform better wired. However, your need for a 20 foot long cable is fairly long and that length will require careful selection and audition (i.e. might pose pairing issues with a headphone amp and the type of headphone you choose) .
With BT you can skip the headphone amp and cabling (for now) while you are figuring out how to move forward.
Check out sites like Headfonia, Headphone.guru, Headphonics, etc. for recommendations and reviews.
B&W just released the PX8 (slightly above your current budget). The previous PX7 S2 is available well within your budget.