I know it's over your budget, but I've wasted a couple grand on shit chairs. I always buy the warranty to get most of my money back, but this is the best chair I've bought, and I'm a bit bigger than you are.
Try something like this. I'm as tall as you, and while I'm nowhere close to the weight rating of this chair, I chose it for the solid construction.
GOAT chair: SPACE Seating Big and Tall AirGrid Back and Padded Mesh Seat, Adjustable Arms, Gunmetal Finish Base Ergonomic Managers Chair, Black
Here's one that supports up to 400 lbs and has lumbar support for a decent price:
There are also some options on WayFair too like this one: https://www.wayfair.ca/furniture/pdp/serta-at-home-executive-chair-sert1280.html?piid=44809703
SPACE Seating Big and Tall AirGrid Back and Padded Mesh Seat, Adjustable Arms, Gunmetal Finish Base Ergonomic Managers Chair, Black https://www.amazon.com/dp/B000WJVB1Q/
I’ve found this site has pretty good options in their other sections and would trust this recommendation, especially after reading the Amazon reviews. http://forbigandheavypeople.com/office-chairs-with-400-lbs-capacity/
So I was looking at options and this thing is on an absurd discount right now if you're in the US:
Sourced from this article which you may find helpful:
I'm a big man. I've had mine for 2 years and it's solid. I left the arms off.
Just got this monster in (once you lift one you’ll understand what I mean), so far worth every penny!
Well if you're happy with it I guess that's what matters, but they do make big and tall office chairs, for example this one has excellent reviews and it's a $800 chair you can get for under $400: