Surly bikes like the Long Haul Trucker/Disc Trucker and the Troll are rated for around 300-330 lbs, but 350 should be just fine as well, as they don't have a hard and fast number for rider weight limit. Both have 26" wheels, for strength, and 36 spoke wheels. Maybe look out for a second hand one within your budget.
Worksman cycles also have a cruiser just over your budget (
https://www.worksmancycles.com/inb-rhc.html).