One way to look at the Constitutionality of a law like this is to pretend that the restriction instead applied to guns. Billed as "common sense safety", the LA law requires abortionists to have admitting privileges at a nearby hospital, notwithstanding the reality that almost all first trimester abortions do not require hospitalization. Pretend that's instead a "common sense public safety" rule intended to make it harder for gun stores to operate, say a rule that the gun store owner be strictly liable for the medical expenses of persons shot with the guns it sells. Most gun stores would be effectively forced to close.
Constitutional? I think you know the answer.
Pro-lifers would do better to enact laws that conform to the duality of interests set forth in Roe v. Wade. A woman must have a meaningful and effective right to choose. But the state can regulate or even ban the procedure following viability, since the woman will have had by that time up to four months to make her choice. Better, then, to pass laws banning abortion after 20 weeks. That fits within Roe v. Wade. But arbitrary procedures intended to shut down abortion clinics in the name of "safety", thereby reducing the entire State of Louisiana to having just 1 to 3 abortion clinics, effectively eviscerates the right to choose.
Just think if there were only 1 - 3 places in the State of Louisiana where you could buy a gun to protect your family!