Former sound engineer here.
The best large diaphragm in your price range is the Sony MDR-7506. It's a stunning headphone for the price. Almost every professional recording studio I know has a couple from that product line on hand.
As others have pointed out, there are a couple of good earbuds in your price range, but this runs against your preference. Earbud sound depends a great deal on fit, and it's harder (==$$$$) to get good bass reproduction on something that small than it is to get it on an overear pair.