Most pretty good 2.4ghz (very low latency) headphones don't support Bluetooth. I think the only one that checks all your boxes would be the Corsair VIRTUOSO RGB Wireless XT. I haven't tried it, so can't speak for or against it. Maybe someone else can chime in. There's also the Steelseries Arctis Nova Pro Wireless, but they aren't nice at all to lug around and have a knack for breaking after a few months.
My personal pick:
A pair of Sony WH-1000XM4 or Bose QC45 and spend the rest on a mic. For gaming or movies, just go wired. Tried and tested, good for traveling, ANC is nice, sound is fine. I can't travel anymore without good ANC after getting used to it, along with all-day comfort.
For the mic itself, remember there's little reason to overspend on a USB condenser. At a certain price they're all limited by USB itself and space constraints - it'll just be paying for a logo and some LEDs. You can argue for software packages like Elgato offers, but much of it can be achieved by free programs such as Voicemeeter Banana. I'd binge some cheap-ish USB mic reviews from someone that has a similar voice as you, if I were you.