ACA = Asymmetric Committal Altitude, the minimum altitude from which an asymmetric approach may be abandoned.
Which part of 'Committal' doesn't the Examiner understand?
As for DA below your ACA, if that's the case then an engine out allowance must surely be added to the DA? On the ME aeroplane I used to fly, we used to add 250 ft to the normal Decision Height for the failure case (so typically 450 ft), with a Visual Committal Height of 350 ft. Once at VCH, you were committed to land and would extend the flaps to the landing configuration. Although that was in a 4-engined aeroplane with a double engine failure, the principles are the same for OEI in a twin.
If the POH states that the AP should not be used during an asymmetric go-around, that's the end of the argument and the Examiner was clearly wrong.
Thankfully I never had to fly a light twin in civilian life, but some of the tales I've heard about Examiners and their approach to asymmetric go-arounds have been...surprising. For example, one 'oral question' was "What would you do if you were below ACA and the runway was suddenly blocked?" To which I suggested the answer "Watch out for the ******g asteroid that's about to hit!". Another asked what you would do in a twin if you had an engine fire which hadn't gone out and you couldn't see the runway at DA....