Great information.
So first the room:
Its fairly big and by 'poor acoustics' I'm going to assume hard surfaces (floor/wall/windows). One thing that often helps in that situation is adding soft furnishings (rugs, fabric seats, bookshelves, acoustic panels) This can remove some reflection/distortion and reveal more detail.
Any speaker will struggle to fulfil its potential in such an untreated room (I had similar experience).
Now the speakers:
From the active monitors you've listed it sounds like you like a neutral sound (not artificial colouring of bass) and you mention you like detail...
One way to go is studio monitors, but as you have an amp something passive may help keep the price down... Also they are designed for nearfield listening so may not shine at more normal listening distances...
Another way to go is with home brands that 'tend' to have a neutral/detailed sound:
e.g. Monitor (older iterations tend towards bright, recent are excellent), Dali, Elac, B&W (some are great but can be a little forward for some tastes), KEF (a little warmer than some others but usually great detail and superb soundstage)...
Of course sound characteristics vary by model, but I would look at these brands (and I have a neutral/detailed taste also). Others on the forum may be able to recommend other brands/models.
Finally, budget. $250 will not buy you 'something high quality' that is new...
However it may be able to get you something 'vintage' / second hand...
e.g.
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I need to caution though that I'm not sure $250 is going to get you a meaningful upgrade on your Mission's... You may get a slightly different sound character that might suit your taste better... BUt you may not and the only way to check is to audition in your room, which means potential returns if you're not happy.
If it was me, I would dress the room in soft furnishings, make sure Mission speaker placement is optimised and then keep saving
Good luck!