Actually Little Yosemite Valley is NEVER closed, well barring a government shutdown of course.
The Happy Isles trail head is in the Valley, so easily accessible year round. The route up along Vernal and Nevada Falls has several variations, they close part of it and keep the rest open as the winter route.
If you are leaving the city by noon, drive fast, and hike hard you just might have enough time to reach the campground in LYV by nightfall on Friday. Can't guarantee that you will be the only one up there, but pretty likely. Very different from the zoo that this place is in the summer.
If you are feeling up for a long hike on Saturday, head up to Clouds Rest, then down the other side and camp at one of the Sunrise Lakes.
Sunday morning hike the short hike down the switchbacks to Tenaya Lake on the Tioga Road. Hitch a ride back to your car in Yosemite Valley. This would only work of course as long as the Tioga road is still open. In snow it would be a long hike back to the Valley from Sunrise lakes. Adjust your plans to the weather, and still be prepared and equipped for snow and cold.