In ASL, may a berserk unit advance in the APh? It seems logical that it can't, but I cannot find the rule.
It can't advance while berserk:
A15.431: ... nor may it move in the APh ...
but, it may lose it's berserk status in the Advancing Fire Phase, and then advance in the Advance Phase:
A15.46 RETURN TO NORMAL: A berserk unit loses its berserk status and returns to Good Order whenever it (or the group it attacks with) eliminates all (but at least one) Known enemy units in its Location with either TPBF (halved) during the AFPh or CC, or if at the end of a charge there is no Known enemy unit in its LOS (15.431).
My memory is that Bezerk units must move including into an enemy hex the maximum movement points during movement phase towards nearest enemy in hexes. Then they get triple fire in Advanced fire phase if they survive Defensive fire.
If the advance during advance phase was removed, then that would make there movement bonus not really a bonus. And sometimes a penalty.
Whatever the current rules say, they should be able to advance, and adding CX penalties is wrong also, being Bezerk, that extra 2 movement points is the bonus of not hesitating.
Not sure what the newer versions of rules might say on the topic.