A link would be helpful.
I don't think you'll be pleased by what you find. It is very clear that DHS had the authority to roll FEMA into this on its own accord.
Does the document describe any criteria for this "override" of standard procedure? And the presence of the term "standard procedure" indicates that the process has variables. What is the "standard procedure?" What is the "special procedure?" Do natural disasters where the state apparatus remains fully functional justify moving from the "standard procedure?"
Katrina was a natural disater, not a terrorist attack, not without some days of warning, and was not an act of war by a foreign power.
It is on the DHS web site. All of your questions are plainly addressed. Their are circumstances in which the Feds can take a 'proactive' approach to responding i.e. doing it on their own. Based on the criteria in the document, there can hardly be any doubt that this qualifies. And the document specifically includes natural disasters within its scope.