There's really nothing good in that price range, especially since you want a DAC also. Only the desktop/near-field Chinese amps are in the territory and they have a lot of pitfalls, not to mention they probably don't have enough power to do the Q350 justice. Andrew Robinson has a good video on cheap Amazon/Aliexpress amps:
Used market may be the best way to go. Check eBay, forum listings, etc. You should be able to find old receivers and/or streamers in the few hundred dollar range.