It could be C. prolifera. But I'd think they would be longer leaved. It could be the flow that makes them shorter.
https://reefs.com/magazine/a-look-at-the-macroalgae-caulerpa/
When you say "it was brought in with the live rock" Where is the live rock from ? Aquaculutured florida, such as tampa bay? If your rocks did come from florida, then further reinforces prolifera theory. They are native to florida waters. They are known to grow quick and successfully outcompete in Florida waters. That's caulerpa in general. They are like weeds!
Is your tank big enough for a yellow tang? Yellow tangs would eat them. Not the best food for them, but they are hardy, so still a viable method of natural pruning. How big is your tank?
Natural predation and manual pruning, lots of water changes, don't dose iron if you do. In order of certainty to less: A baby yellow tang is almost a guarantee, but only if you have future plans with him, rabbitfish eat them too. Don't feed the fish and they will be lawnmowers in the tank. Urchins will eat them and your coralline too: such as L. variegatus, E. lucunter and probably others. Algae blennies, like S fasciatus (hit or miss, i think they prefer micro) , lettuce nudibranch (hit or miss), emerald crab (hit or miss and risky they probably prefer micro), But hey, these are shotgun type recommendations in order.
Maybe try to outcompete them nutrient wise as well. Macroalgae will succumb to the tiny armies, such as those in Vibrant, or carbon dosing. But there could be post treatment bacterial fallout (i.e. cyano outbreak etc.)