Actually the video you had linked tells something very important.
Internal bay does not allow rail-launched missiles to be used without special trapez that extends the launcher so the missile can be safely rail-launched.
Looking at all close in 5th generation missiles IRIS-T, Python-5, ASRAAM, MICA and AIM-9X they are all rail-launched and their integration to stealthy F-35 isn't feasible without fully developing and testing the trapez.
What I do not understand why Lockheed didn't developed such a system in first place clearly understanding that most customers would like to integrate their close-in weapons into F-35's stealth mode - and virtually all close in weapons are rail launched - because as bozon said, in close in scenario LOAL is very problematic.