"I refer to him by, what I assume, is his nickname" vs "I refer to him by, what I assume, his nickname is"

Solution 1:

When nonessential elements are added to a sentence, it should be possible to remove them completely and still have the sentence make good sense.

So the sentence without the nonessential elements would be: I refer to him by his nickname: xxxx.

Then add the nonessential element: I refer to him by, which I assume is, his nickname: xxxx.

Here is more information, if you are interested: Purdue Online Writing Lab and Englishgrammar101

If you don't want to end the nonessential element with "is", you can modify it to read: I refer to him by, under my assumption, his nickname: xxxx.