Does Dark Bastilla’s CBM still apply when she skips a turn

It's almost the same as stun.

In stun, start of turn and end of turn both trigger. For example:
SoT: DoTs effects on character
EoT: -1 to all cooldowns

In stun, character cannot use abilities, assist or evade.

Those are the same in Fear.

Difference comes on +1 to cooldowns on apply Fear and being dispelled on attacking the character.

So yes, it is WAI