> Even being succinct and saying "a representative democracy" doesn't imply a form of government which is constrained by a constitution.

Neither does saying "republic". And "representative democracy" isn't succinct compared to "democracy", its verbose.

> Saying democracy instead of republic just undermines the common understanding of how our government was setup to function.

No more than saying "republic" instead of "democracy". A "federal republic that is also a representative democracy governed under a written constitution" would avoid that problem, but its a lot to write, when "democracy" is the part that is relevant to the point you are trying to make.