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I'm glad I had the forethought to make a copy of the letter, as proof of what had been promised.
According to an ELL post, some people would not like using "as per" as alternatives to "according to". To agree with and support that point of view, I said
As proof, synonyms of "according to" listed on the Cambridge Dictionary does not include "as per".
Is it natural to say that for this particular situation?
I would say that "As proof" is natural, but incorrect here, because it's not technically proof for your side of the argument. The Cambridge Dictionary isn't the faultless guide to English and word choice is flexible.
Starting a sentence with "As proof" is fine and you used it naturally, but it is misleading in this example.
Proof has a specific definition, and I don't think your example uses it correctly. The way you used it is natural in a grammatical sense, but ensure that what you're referencing directly after is actually proof of something. Starting a sentence with it implies that the following statement is technically correct and it's jarring to read something that isn't a proof instead.
