Sorry for not getting back sooner - I forgot about this thread... I've done a couple of different things - either squashed the padding down without disturbing the "fur", or carefully peeled the "fur" back so that I could get at the padding, trimmed or added to fit, then contact cemented the "fur" back in place... If done well enough, you'd think the case was made that way...
Looking at the pics, *I* would use the "peel back, cut, and reglue" method, since the tuner is pretty close to the edge of the case... Contact cement isn't very expensive, and is handy to have around the house... Odds are that the padding under the fur is only styrofoam, FWIW... I'd probably use the "squash" method for the tuners on the other side, if needed - because they don't appear to need much additional clearance...
Hope this helps,
- georgestrings