Please consider that anemones quite possibly live for hundreds and perhaps even thousands of years. They have no complex organs (hearts, livers, etc) to break down over time, there is no evidence to suggest they live anything less than hundreds of years on the reef.
You have a 200g tank that, imo, is already stuffed with fish. Large predators that are both messy and curious. Even if the puffers decide not to sample the E. Quadricolor, the waste they produce will likely be too much for the anemone which requires excellent water conditions and has much less tolerance than even some of the most sensitive fish.
Your Diodon holacanthus grows to 18" in length, your Arothron meleagris to at least that size. These are huge waste producers.
Anyhow, please consider an anemone only tank, void of predators. Here's a quote from WWM on puffers and inverts:
"Puffers for other than fish-only systems? Oh, they love invertebrates- to eat. Even the smallest species will sample corals, anemones, echinoderms, crustaceans et al. to bedevilment; yes, & they'll even eat smaller unaware fishes"
http://www.wetwebmedia.com/puffers.htm
thanks,
Matt