Just cut the pull out and try and get all the runners. I have Palm Caulerpa and I like it. It must not be as invasive as what you have. Cut the runners then get your tweezers and just yank it out of the rock. If you miss a piece just get it when it starts growing. When you cut a stem white stringy threads /stuff with start floating around. I caught the bigger pieces with a net and forgot about the rest. Good Micro Algae for removing phosphates and nitrates.For help to id.
this is fern caulerpa in my display fuge. Also known as caulerpa sertulariodes.
Manual removal and hitting the runners with peroxide or kalk paste for 5 minutes outside of tank and then rinsed with old tank water is best bet unless you have room for very large tangs.
That is Palm Caulerpa. You can just keep it trimmed or move it sump. I’ve read It’s the hardest Caulerpa to usually find and it’s the least likely to go sexual on you. I’d keep it on a island or just trim every now and then. Or throw in sump . It will not over run unless you let it.
You can see all the detritus on the palm Caulerpa. You can vacuum right off the palm or pods/bristle worms and other inhabitants will use it as shelter, a place to eat, sleep, and reproduce. It looks really cool with the wave current in the tank. You can cut it to make it into smaller pieces and glue to a smaller rock. You can remove from tank and dunk it in a bucket of tank or fresh water to clean off. Unless you just can’t stand it I would not remove all of it.