Good bass as in bass-heavy or as in not the typical too weak in-ear bass?
What you pay for is what you get. I recommend sticking with your budget, there are always better, more expensive headphones unless you buy one of those 1000$ custom-fit in-ears (they're about as good as it can get).
For bass heavy I'd say Shure SE215 or DUNU Hephaes. Because of the strong bass they are both a bit weak on the highs.
May I ask on what device you'll be using them?