Skip Joe's Crab Shack off Pelham Road and head straight for Bimini's Oyster Bar, just off I-385 between I-85 and downtown. Casual atmosphere, good food.
Greenville | Bimini's Oyster Bar and Seafood Cafe`
If you want upscale, continue into downtown Greenville and settle in at Rick Erwin's Nantucket Seafood.
Seafood Restaurants in Greenville SC - Fresh Seafood in Downtown Greenville
Not quite as upscale as Nantucket is Ford’s Oyster House and Cajun Kitchen, just over the Reedy River Bridge, near Falls Park and the award-winning Liberty Bridge.
High Street Hospitality Group » Ford’s Oyster House and Cajun Kitchen
One of my favorite places in downtown Greenville is The Trappe Door, a tavern down under Barley's tap Room, just off Main Street. The only regular oyster dish is their Buffalo Fried Oysters, but their mussel pots are simply amazing.
Home - The Trappe Door
All the above are within the Greenville city limits, which means not only that they will be open on Sunday but also that they are allowed to serve adult beverages on Sunday (local option that is not approved in Greenville County outside the Greenville, Mauldin and Simpsonville city limits).
If you want family-style, family-friendly (i.e., alcohol-free) seafood, I've heard that Harbor Inn Seafood on Haywood Road is pretty good.
https://www.harborinnseafood.com/