The AEON dosn't require a lot of juice according to the specs (I haven't heard them) so your choice will likely be down to synergy and source implementation.
The DX200 is a powerful DAP (AM1) and will have no issues at all with the AEON but some say it sounds rather bright and flat, others say it's neutral and revealing.
The X5iii will also drive the AEON with no issue, from a power perspective. I use the ETHER Flow C with the X5iii and they have no issues being driven from the FiiO unit, so I imagine the easier to drive AEON will be no problem as well. Some say the X5iii is veiled and too warm, others say it's musical. I think that AKM's 'Velvet Sound' is an apt description for the X5iii.
The QP1R is the weakest DAP of the three but many find it still able to drive full size headphones just fine. There's the simple interface and scroll wheel to consider as well as a lack of ability to load apps like Tidal because of the simple Linux OS. The QP1R is also said to be on the neutral / bright side compared to some other DAPs. Even though the output power isn't that much compared to other DAPs the Current Mode amplification is said to be very well done.
My point is raw power isn't the only requirement and many people simply think raw max power is all they need. It isn't, and is often the least concern once there is enough for headroom (you'd be surprised just how little headphone's need to reach 90dB SPL). Look for opinions on overall synergy and clean, and well implemented drive for your headphones.
Personally, for audio quality under $1000, I'd say just get a Mojo and be done with it, but then it's a stack you need to deal with.