Murray,
Certainly, your extremely interesting Deep Image shows a dramatically different LMC from the one we can see in short exposures.
Undoubtedly, the bar is way off-centre in the overall stellar distribution of this galaxy.
The pros have made various attempts to image the Mag. Clouds at great depth, for example:
http://adsabs.harvard.edu/full/1991IAUS..148..453I
http://arxiv.org/abs/1008.3727
In our discussion of the structure of the LMC in the recent "observation reports" forum, I presented an image showing the distribution of the red giant stars that define the halo of the LMC;
the bar looks off-centre, but not as far off centre.
Given the complex history of the LMC/SMC/MW system, and ongoing controversies as to how these galaxies evolved and changed with time, it is remarkable that few attempts have been made to really image as deep as possible on the LMC/SMC system.
cheers, Robert