I just want to know whether a Brahmin can shave off his moustache on the occasion of his daughter's engagement or marriage. Are there any other occasions where Brahmins can be clean shaved?
It is not recommended that a Brahmin should have a mustache to start with. So, yes he can shave it off.
See the following verses from the Baudhayana Dharma Sutras 4.5.7:
Now I will proclaim the rites of the Rg-, Yajur-, and Atharva-vedas by which a man will quickly obtain his heart’s desires.
2 He should perform the rites only after purifying his body by the soft recitation of prayers, burnt offerings, oblations, and restraints. He will not attain success otherwise.
3 When he is about to engage in the soft recitation of prayers, make burnt offerings and oblations, and practise restraints, a Brahmin should first get his head and beard shaved and his nails clipped on an auspicious day and under an auspicious lunar mansion during the fortnight
So, specially when a Brahmin is to participate in a ritual, he should shave everything off except the top-knot (Shikha).
In Hinduism, we have the concept of sins being stored in the hair. So, the less hair the better.
Brahmanas are not even supposed to have mustache.
They can have beard, but not a fashion beard (trimmed to certain size, or trimmed only on neck etc.). Basically, whatever grows when you don't shave or trim, let it grow.
and even the reason for letting it grow is because they are not allowed to shave on many days (tuesday, pournami, dvadashi, during asoucha, during wife-pregnancy etc.)
They are supposed to clean shave, including body (above navel) hairs, approx. once a month - meaning no mustache, french beard, goatee, sideburns or any other nonsense :)
They are also not supposed to shave off shikha (tuft / kudumi) at back of head - so no mushroom cut, military cut, mohawk or other more nonsense :)