I understand what you are asking, but the entire sentence is awkward. I would correct it to this:
It happened not long after I had moved to the US.
Which says the same thing more concisely. Let's see if we can come up with a sentence that is less easily recast:
It had been 10 days since the last time we had heard from him.
It had been 10 days since the last time we heard from him.
While the first sentence is more technically correct, owing to tense agreement, the second one would still be common. This is because we often leave out words that are understood.
You should not change your sentence to this:
It happened when it has not been that long since I (had) moved to the US.
In this case, the tenses clearly disagree, whether you put the had in or not. Has not been connects the past to the present, while it happened does not. So, you have a point in time in the past (happened), and are attempting to enclose within it (happened when) a period of time that begins at a point in the past and ends in the present (has not been). That's illogical.