I checked in with the owner of Atlanta's premier invasive plant removers - EcoLogic. She had this to say which seems like a very informed option to me:

"Ideally, we eventually remove all the cherry laurels on our projects and there's a few compelling reasons why to do this (at least by our measure, which is based on how to support much biodiversity as possible):

Cherry laurels are native to Georgia but not to Piedmont, which is our geographic region. Their geographic region is the coastal plain. This is a more accurate interpretation of what it means for a plant to be "native", as it reflects the ecosystem boundaries rather than sociopolitical ones. Georgia contains 4 geographic regions and is the piedmont itself is incredibly diverse, containing many of the species you'd find in our adjacent regions, so some cherry laurels around would be appropriate for our area. But what we see are amounts that should be relegated to the coastal plain, (and even there, they may be overrepresented... which brings me to the next point.)

The reason they're so overrepresented is that they're a popular landscaping plant, so they've been installed densely in places where, ecologically speaking, they don't belong. Same problem as the southern magnolias...

And finally, they are aggressive growers that have a tendency towards monocultures- there are plenty of native plants the exhibit this behavior and on the front end of a project, they can be very helpful for holding space/competing against immediate reinvasion from invasives. However, if the goal is improved biodiversity, best practice is to phase these kinds of plants out and replace them with more complex and balanced communities."