I was waiting for some others to get into the discussion, and @Lou_Ford has thankfully started pointing in the direction I was planning to go. I think the DAP might be a good idea if you’re thinking of a Fiio DAP, as most of them have a bit more oomph than a phone.
I agree with the idea of budgeting something for “electronics” by which we mean DAC/AMP, either in combination or as separates. I note you say that you usually spend time at home doing critical listening. That would argue for something that is not that portable, such as the iFi ( ifi-audio.com) or Schiit www.schiit.com make. If you want something easily portable, consider the iFi Hip Dac, which includes both an amp and a DAC. If you are looking for tiny portable, like so small it becomes part of the phone cord, the Qudelix-5K Bluetooth USB DAC AMP, and the Audioquest Dragonfly Red come to mind.
I think that @Lou_Ford is right, in your price range, you should keep the electronic budget to $200 or less.
I’m not going to “recommend” any headphone. There are users who are more opinionated that will probably choose to do so. I will, however comment… First, one outlet that many use, other than the obvious headphones.com is DROP, formerly MassDrop. They are sort of a buying club, and often get models at very good prices. On Drop, the usual first suggestions are the Sennheiser HD-58x, which is Drop’s model of the HD-580, or the HD-6xx, which is the HD-650. I still listen to my 580 after 25 years. It’s clean and balanced, but won’t be WOW. The 6xx is an interesting animal, and gets into full WOW territory, but we all find that the better the electronics the better the sound. And $200 will give you semi-WOW coming from what you have. The advantage of the 6xx is that is very much a known quantity, and once you listen to that for a while, you will understand reviewers when the refer to it or the HD-650.
Here, I’m going to go off the reservation, but @Lothar_Wolf and several others will agree with me. Based on what you have said about your sound preference, you might try GRADO. I have a couple of them. Headphones.com sells them and they have some open-box deals. If you don’t like the foam pads - many don’t - Dekoni and Beautiful Sound have nice alternative pads. One of my faves is the RS1e with Beautiful Sound pads. If you try GRADO, then search “Tape Mod” …
Another reliable choice is the Hifiman Sundara. This is a Planar-Magnetic design, and is available from the source that owns this forum (nudge nudge, wink wink). Also they sometimes have open box discounts. This headphone will benefit from some electronics. And iFi offers a signature combination in your price range that is designed for the low-end Hifiman.
That’s all from me for now. I think there’s no excuse for those other Fora and Pharta that are rude to people on budgets. I’m a proponent of the 95% solution (which thread you may search here). Because that last few percent costs a bundle. And many of us have other things to do with our money.