Personally, I don't like to tinker with the words of time-honored beloved songs like this (unless a word is really offensive in today's society). And I take "brotherhood" as just as gender-neutral as is the term "fraternal twins."

But if want to keep the God reference gender neutral, one simple way is to change the phrase,
"...God shed His grace on thee," to
"...May God shed grace on thee."

This seems a very minor change and does not involve coining new terms or using awkward phrasing.