Had big problems refreshing an inline CSS style on my site having ModPageSpeed enabled. Usually it is as easy as bumping the ?ver=1.2.3 in the stylesheet URL but wasn't working. I guess since it was inlined, other rules apply?
In any case this one liner will trigger the flushing, at least on my setup:
touch /var/cache/mod_pagespeed/cache.flushCode language: JavaScript (javascript)