Help me choose headphones for music / TV

Hey guys,
I bought a Fischer Audio FA-003 almost a decade ago, been using that ever since. It’s a very neutral sounding headphone and I need something with a bit more bass, not a sony bassy headphone but something more than this has. I would use it for music mainly and movies, I want to watch Joker at home with a proper headphone that can give me back those deep notes, especially from the cello/violin (I think it’s cello), that made me go back to see it the 2nd time honestly.
Now I’d like to not spend 500$++ if possible but as far as I see there are 2 options:

  1. get a bluetooth headphone that can connect to the TV directly without any dac/amp, I only ever tried one BT headphone, MDR-1000XB but I didn’t like it because it was very weak on BT, with cable connection it sounded much better but still not even as good as my FA003, so if all wireless BT headphones are similar (within that price range, 250$) then BT is not even an option
  2. Buy a dac and amp, best value for this would be the Topping DX3 Pro (even tho I’m scared from that 200+ page thread that is full with problems of the dx3) or anything else would be a lot more expensive, like atom + d50s, almost double price (atom is basically 160$ here, eastern EU) and since nothing else in this price range has a remote control (need it for the TV), it’s dx3 pro and a headphone. I don’t know which headphone would be good, I realize I don’t need a bassy headphone because they are just too bad for anything other than bass heavy sound. Something warm/dark but not too bassy, something that brings out instruments like cello/violin and it’s not too bright.

I’m hoping someone will say that MDR-1000XB is bad and other BT headphones are much better because that would save me buying a dac/amp but I expect that BT headphones are just weaker in general.

I wouldn’t suggest a bt headphone because of latency issues (and the sound quality drop tbh)

So you like a neutral headphone with a bit more bass? Are you in the market for an open back or closed back? And how much are you willing to max out at for your budget for a headphone + a dac/amp

I only have the FA003 closed back so I never listened to an open back headphone. I’d like a headphone to be no more than 300$-350$. The dac/amp will be dx3 pro (220$) or maybe something else but I really didn’t find anything with a remote and better price/quality (MX3 is 130$ but that has a substantially worse headphone amp and worse dac), so far now I’ll get the dx3 and maybe later get a SMSL SP200 THX amp with a Topping D50s dac ? That’s kinda endgame for me.

I don’t really know how good are open back headphones, would sound leak in and out very much? Like the neighbour’s dogs, would I hear them barking while listening to a movie at 80db? Because that would suck a lot. If not and if open backs can provide sufficient bass, then maybe I’d like to try one out. I read the soundstage is vastly superior to closed backs and that I’d like.