Ohio is almost setup exactly like Louisiana in that regard. The Ohio State University is the state Flagship, is head of a single Board and is a BCS member of the Big 10. Yet, Ohio University also exists as a member of the MAC along with Kent State and Miami Ohio. The other university is Cincinnati which has now worked its way up to the Big East. Other private schools include Dayton and Case Western. Miami University in 1986 and Ohio State in 2001 were noted as "public ivy" institutions, the state/public equivalent of a private Ivy League institutions. Ohio University's journalism school ranks, with Columbia University's journalism school, as one of the nation's best. Ohio State and Ohio University certainly co-exist nicely along with other accomplished universities and there is no question which is both the Flagship and BCS program.
Why can't that exist in Louisiana? LSU is the Flagship and only BCS school in the SEC. UL, ULM and LA Tech exist in the SBC and WAC, much like the MAC. Tulane is in C-USA. There is absolutely no reason the University of Louisiana cannot exist in Louisiana without detracting from LSU's status as the 1st Tier school. Just ask The Ohio State University how that is working out for them. There is no reason for LSU to fight this.
Likewise, there is no identity problem with the names. In Texas, there is Texas, Texas A&M, Texas State, Texas Tech and UTEP, UTA, and UTSA. Am I leaving any out? Texas A&M constantly refers to itself as the Texas Aggies, dropping the A&M. Still no problem. There is no confusing, LSU, Louisiana, Louisiana Tech and ULM. Its too bad ULM doesn't go the Louisiana A&M route to solve the name issue. Then, we would be almost identical to Texas. No one confuses any of those schools.