Funny Derek,
I think you are right on. The "However" applies to the sentence previous that birthdays are in a bad light in the Bible. The portion of the next sentence in commas is the rule for deciding if one could use their conscience.
For example, the Bible definitely puts birthday celebrations in a bad light. However, if it is very obvious that a custom has no current false religious significance and involves no violation of Bible principles, each Christian must make a personal decision as to whether he will follow such a custom.
So if it were obvious that "Birthdays" have no CURRENT false religious significance and violate no Biblical principles, which they don't, then one could make a personal decision.
That is allowed by the grammar of the sentences. If the WTS really cared why would they use a phrase like "bad light"? What kind of a phrase is that? What the heck do they mean? That could mean anything. Shouldn't a religious organization who claims to be responsible teach to all mankind about God's requirements try a little harder to articulate what they mean? I agree with you that they ambiguity must be deliberate or else they are really losing their academic abilities.
They should clarify whether or not "bad light" is equal to " false religious significance and violation of Bible principles". Since they do not it would be wrong to assume that is what they mean.
The two sentences clearly go together as an "example" using birthdays as the subject. So when the second sentence uses custom one could understand that to mean birthdays.
It could have been written better.