"Can a non-fatiguing system be deployed in a brighter-than-ideal room and have it remain not-fatiguing?"
The short answer is, "yes". But that's not very interesting, so you're going to get the long answer too.
Let's look at a couple of causes of fatigue that relate to room acoustics, and then come back to your question.
A significant spectral discrepancy between the first-arrival and reverberant energy can be a cause of listening fatigue. Sorry but I'll have to delve into psychoacoustics a bit at this point.
The ear localizes sound sources primarily by the first .68 milliseconds of a sound impulse. This corresponds to the time it takes for a sound wave to travel the roughly 9 inches around the head from one ear to the other. After .68 milliseconds, something called the "precedence effect" or "Haas effect" (after Helumt Haas, who first described it) kicks in and the ear ignores repetitions of the original signal (reflections) as far as directional cues go. After about 40 millisecond or so (I forget the exact figure) the precedence effect wears off, and a reflection is heard as a distinct echo.
Now how the ear does this is very interesting: It takes each new incoming sound and puts it into a short-term memory for the duration of the precedence effect. Any and all incoming sounds are compared to all of the sounds in this short-term memory to see whether they are "new" sounds or "old" sounds - and if the latter, they are classified as a reflection and ignored as far as directional cues go.
What do you suppose is the criteria the ear uses to determine whether an incoming sound is a repetition of a sound that's already stored? It compares their spectral content! Very simply, it looks at whether the incoming sound "sounds" like one that is in the short-term memory. This is where the fatigue factor comes in: When there is a substantial spectral discrepancy between an incoming reflection and the stored original, it is literally more difficult for the ear/brain system to correctly classify that incoming reflection. More CPU usage, and so the CPU heats up. After about half an hour, listening fatigue can set in.
On the other hand where there is only a minor spectral discrepancy, and preferably where said minor discrepancy is caused by the room's characteristic acoustic signature, listening fatigue does not arise, at least not from this cause. I am not saying that a spectral discrepancy between the direct and reverberant energy is the only cause of listening fatigue, but I believe that in many cases it is a primary one.
The way to quickly check the spectral balance of the reverberant field is to walk out of the room and listen through an open doorway. All you can possibly hear from outside the room is the reverberant field. If it still sounds realistic, then the reverberant field has a good spectral balance.
Let's briefly go back to what happens between .68 milliseconds and the fading of the precedence effect. Although directional cues are suppressed, loudness cues are still picked up (singing in the shower as an example). So if there's a lot of extra reverberant field energy around, say 2-5 kHz (not uncommon with a typical small two-way system), the ear picks up on it and that region sounds louder than the rest of the spectrum, even though it might measure perfectly flat on-axis! Which brings up another potential source of fatigue: Excess energy in the 2-5 kHz region. The ear's sensitivity peaks at about 4 kHz, and many speaker systems have excess off-axis energy in this region because of the tweeter's wide radiation pattern at the lower end of its passband.
One elegant solution to both of these issues is for the speaker's radiation pattern to remain uniform over as much of the spectrum as is practical. How to do that is another topic for another day.
Okay, let's look at the question again:
"Can a non-fatiguing system be deployed in a brighter-than-ideal room and have it remain not-fatiguing?"
A brighter room simply means that the reflections are full-spectrum instead of rolled off at the top end. If the reverberant sound generated by a speaker sounds bad, then you probably want to absorb as much of it as possible. If the reverberant sound from a speaker sounds good, then you can preserve it and have a fine-sounding, non-fatiguing system. And now you know why.
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