I know that Christ Church's swim team uses the pool at the LifeCenter across from Greenville Hospital for practice. I'm afraid that I don't know much about public schools and swim teams, JL Mann has swimming and has won state championships but I don't know where they swim, Wren does not appear to have swimming so I'm not sure if Powdersville will when it opens and Riverside has swim team but I don't have any info. Alot of the neighborhoods are involved with SAIL which you can get more info at
www.swimsail.org. I'm sure you saw the information on Team Greenville through Westside Aquatics. Also even if you do not live in neighborhood that participates in SAIL the CC's do and many of the neighborhood pools are open to outside membership such as Stone Lake, Gower and Sugar Creek to name a few. I hope this helps.
Also when my kids were taking swim lessons at the LifeCenter I know the director was talking to another family about the competitive options options in town so you may find more information by talking to the aquatics programs at the LifeCenter or the Caine Halter Y.