GOAT chair: SPACE Seating Big and Tall AirGrid Back and Padded Mesh Seat, Adjustable Arms, Gunmetal Finish Base Ergonomic Managers Chair, Black
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.
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:
https://www.amazon.com/gp/product/B000WJVB1Q
Sourced from this article which you may find helpful:
http://forbigandheavypeople.com/ergonomic-office-chairs-for-heavy-people/
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!
https://www.amazon.com/Space-Seating-Adjustable-Gunmetal-Ergonomic/dp/B000WJVB1Q
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: