Sam's in Everett is open 9am-9pm Mon-Sun, but that's not exactly S. Seattle.
I used to live in S. Park on the Burien border, so I'm familiar with the area, but I don't know of any ranges down that way.
There's also the 100 meter (300 meter for members) range in Kenmore, just up over the hill from the intersection of I-405 and the Bothell-Everett Hwy. (St. Rt. 527).
For a decently comprehensive list, check out the NRAHQ list: Where to Shoot (WA)