... displacement. The problem, as it seems, these may also be non-linear. To this you also have standing waves within the cabinet which add to non-linearities.
Two forms of "non-linear", namely by amplitude (frequency response) and harmonic distortion (frequencies created, in lack of a better word). These two forms are different in origin and effect.
All of these cabinet vibrations or distortion should ideally be measured in the direction to driver force. Not at side walls.
Depends - alas the above mentioned link is only available in German. I may summarize the findings:
Yes, to use CLD can help to a degree,
but bracing, even of a simple kind helps way more,
while the panels' resonances are driven to the major part by the internal air volume's resonances,
so that bracing does not shift a panels' observed resonance frequency,
and additionally no break-up modes are expected,
which should all be set into critical relation to sound escaping through the driver's cone alone -
not the least harmonic distortion from the panels alone was utterly neglegible.
The author experienced some severe trouble with glueing the CLD to regular panels made from MDF. It separated within weeks in the hide-out of a speaker's enclosure, unobserved.
Once the tech/ parameters are understood, your problem seems to originate in a not-so optimal construction of the type MDF box. Alas, it would be very problematic to investigate the case of the casing ;-) after the fact remotely.