As we informed you by phone, we shall be starting our marketing campaign in Asia from next month.
Your "as informed over/on/by phone" is ungrammatical, since it can only refer to "we". It seems unlikely that you would be telling somebody that you have informed yourself in a phone conversation. In addition, if you wanted to use "over" or "on", it would have to be "the phone" - but "by phone" is the normal phrase.
I suggest that you read about "dangling participles" (link to University of Bristol) - "informed" in your sentence is a dangling participle.