I'm in a square room which isn't ideal but have found that other brands sound fine. Is the "bloom" likely to be fixed by some kind of room treatment?
Yep, the technical term for that is 'ringing', as these large dipoles (radiate equally front and rear) will cause modal ringing at frequencies dictated by room size, speaker placement, and room surface materials.
A square toom is indeed the worst shape from a modal perspective, although that is usually in the low-end, but will indeed affect midrange, as you report.
Yes, room treatments will easily fix the midrange and up issues.
I'd first start by reducing the rear-wave energy by placing absorbers (4" non-reflective, RealTraps MiniTraps HF or equivalent) offset from the wall behind by 2 to 3" such that they absorb (coefficient of 0.9 or better) from 300Hz on up. Place them in the line of fire from the rear of the panel, so if they are toed-in, the panels on the front wall will not be directly behind the speakers, but slightly outboard, centered with the direction the rear of the panel points at.
The next place to add treatments to mitigate that ringing will be the rear wall (behind the listener), so I'd also use non-reflective treatments 3 to 4" on the rear wall, covering the entire area behind the listening positions. Generally, an area 4' tall by 6 or 8' wide. post a drawing (hand-drawn is fine) of the room, the perforations (doors, windows, etc.), measurements, and locations of the speakers.
Since it's a cube, I'd also add some lateral absorption to mitigate ringing in that axis. So another pair of 2x4' 4" absorbers on the side walls behind the speakers (close to the front wall) is positioned to mitigate whatever 'bounce' from the rear wave might be coming off the front wall. Think billiards to find the right placement.
I'd also place some absorption on the side walls opposite the direction of travel from the front of the panels. In a cube, guessing that will be near the rear corners. If aiming at the corner, then the treatment would straddle the corner.
Do you have subs as well?