He was called Big Joe.
Big Joe, as he was called,...
He was known as Big Joe.
Big Joe, as he was known (as?),...
When I was little, I was taught that no one with an ear would say "as he was known as", and no one with a brain would say "as he was known"; the way out was to retreat to "called". It is hard, tho, to keep a slippery little word like "as" in any kind of grammatical cage, and these days "as he was known" is all I ever see in print.

Now here is someone in the New York Review trying the same trick with "referred to" & making a mess:
Soon, however, Israel had its sharpest clash in years with the United States over the "delicate subject," as the Israeli press continued to refer to the new nuclear reactor.
On first reading this, I took "as" to be a conjunction = "during which time", and supposed that the references (cryptic or not) continued into the crisis & perhaps contributed to it. I didn't see why that would matter so much, but what do I know about diplomacy? Then it dawned on me that what the author actually meant was

Soon, however, Israel had its sharpest clash in years with the United States over the "delicate subject" (which is how the Israeli press habitually referred to the new nuclear reactor).

As the press continued to refer to the reactor as, you see. The author made the false scent a good deal worse by writing "continued to refer to" instead of "continually referred to". At a guess, he first thought "kept referring to", wanted something more formal, and failed to notice that "kept" is ambiguous & he had chosen the wrong translation.

Joe Fineman (Email Removed)
He was called Big Joe. Big Joe, as he was called,... He was known as Big Joe. Big Joe, as ... to", wanted something more formal, and failed to notice that "kept" is ambiguous & he had chosen the wrong translation.

Yes, wholeheartedly, at all points; but I have a nasty feeling I might be unwise to cast even the second stone!
Mike.
He was called Big Joe. Big Joe, as he was called,... He was known as Big Joe. Big Joe, as ... in any kind of grammatical cage, and these days "as he was known" is all I ever see in print.

Although I see your point, since "know" and "call" have different meanings I think you and I should both learn to live with "as he was known (as)".
Now here is someone in the New York Review trying the same trick with "referred to" & making a mess: ... press habitually referred to the new nuclear reactor). As the press continued to refer to the reactor as, you see.

Yes, I see. But as you've already demonstrated, there's little out of the ordinary about this construction. I'm sorry you tripped up on it, but I didn't.
The author made the false scent a good deal worse by writing "continued to refer to" instead of "continually referred ... to", wanted something more formal, and failed to notice that "kept" is ambiguous & he had chosen the wrong translation.

A unlikely scenario.
Adrian