I don't see a weakness that is common to all generated undead, but you can make your own, and either make them weak against a material that your preist character shoots out or change their class from general_poison to undead, and have you preist character apply a syndrome, with line of sight, that only affects them, and kills them like a basalisk stare.
[INTERACTION:SPECIAL_DEATH_CURSE_1] - in an interaction file
[I_SOURCE:CREATURE_ACTION]
[I_TARGET:A:CREATURE]
[IT_LOCATION:CONTEXT_CREATURE]
[IT_CANNOT_TARGET_IF_ALREADY_AFFECTED]
[IT_MANUAL_INPUT:victim]
[I_EFFECT:ADD_SYNDROME]
[IE_TARGET:A]
[IE_IMMEDIATE]
[IE_ARENA_NAME:Paralysed]
[SYNDROME]
[SYN_NAME:basalisk stare]
[SYN_AFFECTED_CLASS:UNDEAD] -- for just a basalisk change this back to GENERAL_POISON
[SYN_IMMUNE_CREATURE:BASALISK:ALL]
[CE_PARALYSIS:SEV:100:PROB:100:RESISTABLE:SIZE_DILUTES:START:5:PEAK:10:END:20]
[CE_SKILL_ROLL_ADJUST:PERC:0:PERC_ON:20:START:0]
[CDI:WAIT_PERIOD:30]
In the creature or a class similar to necromancer
[CAN_DO_INTERACTION:SPECIAL_DEATH_CURSE_1]
[CDI:ADV_NAME:Stare]
[CDI:VERB:stare:stares:eyes lock]
[CDI:TARGET:A:LINE_OF_SIGHT]
[CDI:TARGET_RANGE:A:25]
[CDI:USAGE_HINT:MAJOR_CURSE]
[CDI:MAX_TARGET_NUMBER:A:1]
[CDI:TARGET_VERB:turning stiff:turns stiff]
[CDI:WAIT_PERIOD:30]