I think the author explains pretty well why this is unlikely, without calling the pilot's claim BS. The F-117 didn't have a radar, so could not use the radar guided missiles of that time, and opening the internal bay and somehow lowering the sidewinder enough to achieve IR lock before firing seems pretty sketchy. A Lock After Launch missile would be needed, which we didn't have yet.
After being called out, the pilot recanted his claim:
In the end, Donaldson clarified his statements on the topic and to discuss the matter in a bit more detail with The War Zone. In short, the F-117 never had the capability nor has any missile ever been fired, not even for testing. That’s perfectly understandable. Much less understandable is the reason why the pilot claimed it in the episode in first place and also confirmed it in another episode Q/A session.
Lockheed developed a radar that sat behind a sliding door where the aircraft would take "snapshots" instead of leaving the radar on and exposed. The door would quickly slide open and shut, giving away minimal signature while the image was snapped and then sent to the missile.
Single targets would be an easy adjustment for the missile. Picking one out of multiple targets could be adjusted by the program in the missile circuitry after it launched.