I must agree with Globetrotterjon, vintage is a great option, but take a little time to research the pen to see how much it is worth, and whether the pen is in working order etc.
Noodler's pens are not only for tinkerers... I got my Ahab in perfect working order, no tinkering necessary!
I agree about the Noodler's. My Ahab worked just fine upon arrival. It needed a couple days of flexing the nib to make it easier to do said flexing, but now it's a flex-master!
And I also agree about vintage pens being a good place to look for flex, but none that I know of fall that fall into the budget, aside from cheap, cheap "junkers," and even then, it's hit or miss with the nibs. Some might flex, some might not.
For modern pens that I know of closer to your budget, my Kaweco Classic Sport in fine has a bit of flex. Not much, but it goes from fine to bold with a little pressure. Some Parker Vectors have a bit of flex as well without breaking the bank. Neither of these are considered to be "flex pens," though. And of course, Noodler's. I would say to try one, but I might be a bit biased, haha. And if you do end up needing to tinker, they're easy to pull apart and put back together, so it shouldn't be that much of a fiddly experience.
Unfortunately, I don't know of any Asian pens with flex. All the ones I own are hard as nails. But I'll have to look into the Serwex myself!