Water at Lodgepole would be reliable.
Here is Hikin’ Jim’s Caltopo. Mine Shaft Flat is #23 in the left pane. Zoom in to find Mine Shaft Flat Campground
It is off the North Fork Meadows trail as you suspected, but your assumption of .3 to Lodgepole is not correct. From Fish Creek Saddle, it would be approx .3 miles to Lodgepole, but from Mine Shaft Flat, you’d have to backtrack to FC Saddle and then down to Lodgepole
On the map, there appears to be a water source farther past MS Flat labeled Mineshaft Spring/ North Fork Spring. I’ve never been down that way so hopefully someone will be along to let you know if those are reliable water sources or not.