Isaac, I gave a part once (local needs) where I got to tackle that issue directly.
I explained that just as the declaring of something as "corban" , "or a gift dedicated to God" didn't outweigh the prior responsibility one had to take care of his or her parents, so too, the declaring something to be a "conscience matter" didn't remove the need or responsibility to be able to "explain the reasons for the hope in you [as to your belief in the Christ, that he rose from the dead, or any of the other things a christian might profess even as "a matter of conscience"] with a mild temper and deep respect."
I said there ought be a scriptural foundation for any deviation from the base, or natural reading and application of a scripture. In "matters of conscience", the matter is a matter whereby between ones thoughts "he is either accused or excused". If he is exonerated in his own mind, then the exposition of these reasons should be able to find their expression in the language of scripture. Otherwise, if there is no scriptural foundation, then "conscience" has nothing to do with it - it is a matter of "taste, or preference" and not "conscience" at all.