If someone thinks I'm faking my identity, trying to be someone I'm not, and claiming me to be "A" can I use:
He suspects me as A.
Is the use of "suspect as" natural?
Your usage of "suspects" is correct, but it would be better for you to say:
He suspects that I am A.
He suspects me of being A.
Saying "suspects me as" could be a little unclear because it can also mean that he is doing the suspecting and he (not you) is A. In other words, "As A, he suspects me." But since that is not your intent, it's best to avoid the ambiguity.