ULGrad@HOU and I made the trip in 1984 to the state capital as well. Knowing the history and circumstances of the name are important. We did get the name changed to "UL" by the people in charge of doing so at the time. That is our claim to "UL". It was done and it was legal. That situation did not happen to ULM or anyone else... it was our legacy and despite having processed and accepted "UL-Lafayette" as the legal name, we see no reason to use the complete name as many other universities use shorter versions of their official names when they desire to do so.
"We" did not change our name. The governing system we applied to granted our name change. Those who did not like us having the name UL (granted due to academic success and size of the university comparatively) leaned on their state representatives who changed the law and rescended the name change by retroactive application of the new law. And it wasn't a surprise to me that we did not win the state lawsuit.
When the UL System and naming conventions were created, we went thru the required steps to get our official name changed to U of L at L. We abide by the legal name where it explicitly states we must... but the reason for sticking to the "UL" shortened version when and where we can, is perfectly understandable when you know the history. We will stick to the shortened version and it does not matter if there are conflicts at times.
If we get our athletic system act together... it would make it much easier to clear up our name request in regional and national circles. Other than those passionately connected to other Louisiana institutions, if we achieve sustained major sports athletic success, there are few reasons that "UL" will not become much more common on regional and national coverage opportunities (we just need to capitalize on those opportunities).