I would also consider RE0 which IMO sounds at least as good as Q-jays, UE700 and DBA-02 at ~1/2 their price. RE0 sound fine unamped, but adding an amp later will improve their sound significantly, which cannot be said about Q-jays, UE700, DBA-02 and most other IEMs for that matter. Most other IEMs hardly improve with better amplification, if at all from my experience.
Also, you can find a used Ortofon e-Q5 or e-Q7 in the for sale section of the forum for $200 or less in excellent condition. These are moving armature IEMs that sound a cut above any dual driver BA I tried. Personally, if I had a decent amp I still prefer the RE0 slightly over the e-Q5/e-Q7, but unamped the latter two will be significantly more enjoyable for most people. I would personally go with the e-Q5 over e-Q7 if you are looking for the most balanced sound. e-Q7 is more mid centric and has a treble roll off compared to e-Q5, but can sound more musical depending on what you listen to and your tastes. I think e-Q5 has the more advanced driver. Compared to dual BAs, such as Q-jays or DBA-02, the e-Q5/e-Q7 have better dynamic range, tighter, more textured bass, better texture in the mids and highs, a more realistic timbre, better overall tonal balance, as well as more richness and weight in the sound. The BAs still win in separation, clarity and speed, but the difference is fairly minor and overall I feel that the moving armature drivers come close enough to BAs in all areas, while clearly beating them where it matters most, like in timbre, texture and richness.
For $200 or less you may also be able to find a used RE272 on here. RE272 IMO, is the best sounding IEM of the ones I mentioned above, although whether or not you will find it more enjoyable depends on your personal tastes. Technically, RE272 offers better clarity and speed than moving armatures and I think pretty much matches balanced aramture drivers in these characteristics. RE272 also offers the advantages of dynamic drivers, such as richness, texture, smoothness and realistic timbre that is just not there in BA drivers. At the same time it matches or surpasses most BAs, even the triple and quad driver ones in all other ways IMO. Overall, RE272 is a really winner technically to my ears, but it does sound quite analytical and balanced and will not suit those looking for a specific pleasant coloration. RE272 has no colorations and just presents what's there in your source and music, but it is also very sensitive to tips and fit so it may take quite a bit of experimenting and patience to get the most out of these IEMs. In this regard, the moving armatures like e-Q5 and e-Q7 are better because they sound great straight out of the box with the stock tips. But if you are willing to take your time and experiment, the RE272 IMO is better.