I think we worry too much about that. Before our successes the past few years, and back when the name was new and it was not clear who was who, it was important.
But the additions of Austin, Berkeley, Champaign-Urbana, Athens, Chapel Hill, etc., don't hurt those schools. In fact, the location name makes it clear that there are multiple campuses in their system.
We have established ourselves. We are the largest, most comprehensive, and foremost research campus in the system, with the most success in athletics. Inside of Louisiana, everyone knows who we are are.
Outside of the state, the 'at Lafayette' and second 'L' are quickly disappearing. So the rest of the world knows who we are, too.
Complaining about it just gives aid and comfort to the enemy. We have plenty of mountains left to climb, so let's stop focusing on the ones we have already conquered.