This is an interesting question. Both the Merriam-Webster and the Cambridge dictionaries list the words “as” and “per” as prepositions. It is not unheard of to place two prepositions together to create a compound preposition. Examples of this would be “instead of” and “according to.” We might even put three together. Think about “on account of.”
When we look to see if these two prepositions can be used together, it comes down to a difference between British and American English. The Cambridge dictionary lists “as per” as only an idiom. We’ll talk more about that concept in a moment. The Merriam-Webster dictionary lists “as per” as a compound preposition.
Thus, if you want to be technically correct in British English, you would write “per our discussion” or “per your instructions.” If you are writing for an American audience you may choose to use “as per our discussion” or “as per your instructions.”
The fact that “as per” is an idiom in British English tells us that in the past it may have been an idiom in American English. American English began life as British English and has evolved over time. Some idioms have now become accepted phrases.
This can clearly be seen in the fact that when Noah Webster wrote his dictionary in 1828, he included 70,000 entries for American English. Contrast that with the 470,000 entries found in a modern Merriam-Webster dictionary and you can see that words and phrases are added to a language over time.
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