Hm, that doesn't look extremely "peaky" (it's more
.ı|ıı|ı|ıı. than
..|....|.|..), but still I wonder if the reason the volume of your export is lower may be dynamic range.
As far as I can see, the peak circled in red is the only one touching the "ceiling": that one is keeping all the rest down. If it was a little shorter, and the other peaks too, the music
between the peaks could be louder.
What determines a track's "loudness" is its
dynamic range, meaning: the difference between its loudest peak and its average volume. If everything is too quiet, but
equally quiet, no problem: you can just turn up the gain for the whole track – which is trivial in digital audio, since there's no analog noise that gets turned up with it. But if your track has peaks that are much louder than the rest, those peaks will hit the ceiling first and keep all the rest down.
The ceiling meaning: the maximum volume for a given format. It's higher in 24-bit than in 16-bit audio, but still something to take into account. Plus there's the equipment you'll be playing it on, which also has its operational limits.
But why the difference with playback in Live then?
Well, when you're playing back in Live, you can ignore the ceiling to a certain extent: Live works with a high bit depth internally, and you may not even notice some peaks clipping (being cut off by the ceiling of any part of your audio setup) if they are very short. That can give a false sense that your mix is balanced. But as soon as you export, even if there's just one errant peak that's much higher than the rest, that ceiling comes into play.
The Normalize option is meant to keep your exported audio within the clipping ceiling, yet as close as possible to it. However, all it can do is apply positive or negative gain equally
for the whole track, meaning all ratios between loud and quiet are preserved. So if your track has large differences between loud and quiet, all Normalize can do is turn
everything down until those peaks fit under the ceiling.
The solution – apart from identifying the instrument causing the peaks and turning it down in the mix, which also decreases that instrument's average volume – is limiting: a limiter can rein in the volume specifically above a certain threshold instead of for the whole track at once. It will catch those peaks and bring them closer down to the rest of your track, to stop them hogging the ceiling and give everything a chance to be heard.
Give it a try with Live's built-in limiter on the master track. Set the input gain so that the gain reduction meter (the one that works "upside down") is only triggered on those peaks, and see if the export sounds more like what you expect then.
I could be entirely wrong of course! Maybe it's not about dynamic range, maybe phase cancellation does play a part here. I'm still not clear on why the violins are so different in stereo and mono. Are they playing two different parts or exactly the same?