Retired English Teacher
As to 'beggar the question', I googled that - and found very few examples, while googling 'beg the question' brought up many. A search on the more relevant COCA for 'beg/begs/begging the question' brought up 49/276/25 examples. My search for 'beggar/beggars/beggared the question' brought up 0/0/0 examples.
This would suggest that the natural idiom for native speakers contains the word 'beg', not 'beggar'. I don't deny that some people might use 'beggar'. Some people can mangle any expression. It might even have been common in your school, possibly as a joke for those in the know.