This happens to most if not all of the "missile-only" equipped fighters.
The reason for such planes' AI not engaging enemy fighters is within the internals of AI logic.
That logic simply says "If you've got no ammo left on trigger 1 and trigger 2 (guns & cannons), then you cannot engage enemy aircraft".
In Ultrapack we've got one such plane (MiG-17PF) and to overcome that issue, we've implemented a "fake" gun that carries a dozen bullets as long as there are missiles left (with auto-reload on every bullet spent) and 0 if not.
Works a treat.
Unfortunately, the creators of most (if not all) jet fighter mods didn't take this issue into account.
You can't blame BAT for this as BAT is merely a pack of existing mods than a new mod of its own.
Another issue with missiles of fast jets is that AI has no logic implemented whatsoever that would enable it to get into a promising shooting solution for the missiles it carries.
AI will always attempt to get into a gun-shooting solution.
Missiles get launched by a separate logic that simply checks whether or not the current solution falls within the missile's engagement envelope. AI's maneuvers don't care about what that envelope looks like.
As a result, short-range missiles with a wide envelope (all-aspect AIM-9s for instance) are much more likely to see AI launching them than mid/long-range missiles with a narrow envelope (early Sparrows for instance).
The solution to this would be a completely rewritten AI logic.
Go ahead, volunteers!
Mike