Keep in mind Tulane juices up applicant #s significantly by charging $0 app fee and relentless marketing to just about anyone who opts in for mailings with college board.
So yes, it is a competitive school, but I have been told there are a lot of lower GPA/score students who apply anyway in the regular decision round (and are rejected) due to it being on Common App and free. Once you have done all the work to apply to other common app schools, why not give it a shot with zero application cost?
My hunch is your honors and merit qualified student will be just fine…