Is there such a thing as "the wrapper fallacy"?

Basically, this "wrapper fallacy" is when someone picks a hypothesis that, despite appearing simple, is more complicated than the problem it corresponds to. In other words, it is a deceptively simple hypothesis.

By the conjunction effect, the more complex a hypothesis, the less probable it is and the more evidence that is required in its favour. This applies to "wrapper" hypotheses which entail many more assumptions than they might suggest at first glance. Favouring such a hypothesis over others, without the requisite evidence, would fit nicely with the least plausible hypothesis fallacy.

An often missed part of Occam's Razor is that it is simplicity only matters "all other things being equal." This includes probability. We have billions and billions of examples of natural events, and zero examples of supernatural ones* (only stories of supernatural ones). So even if God or God's magic is the most simple thing in existence, Occam's Razor does not apply because all things aren't equal (i.e., we need to factor in probability).

In short, I would say what you are calling a "wrapper fallacy" is more of a misunderstanding of Occam's Razor.

* It can be argued that demanding evidence for the supernatural is an impossible ask, and I would likely agree. But it is not the job of someone who does not believe in the supernatural to explain to someone how the supernatural can be demonstrated. Even those who believe in the supernatural must admit that anyone can claim "supernatural" for any reason - from the birth of a universe to finding one's car keys - so it is horrible, unfalsifiable, "methodology."

In addition to what Bo says, when you ask: “The universe exists. What is its origin?” you are starting too far up the hierarchical ladder of logic. The question assumes an unproven and even an undefined premise: That the universe needs a cause. If it has always existed, it never had an origin. You must resolve that before you may logically continue the discussion. 

If you can resolve that issue, then you have the problem of defining god in the context here. So, restated: precisely what is this god?

Logically, in any discussion, one must present (1) an intelligible definition of the subject under discussion [in this case, god and the universe] and (2) adduce evidence to support the proposition [in this case, the existence of the god defined]. But unfortunately, no one has ever presented me with (1), so we never got to (2). 

To me, the problem faced by those who profess a belief in a god — any god — is not they cannot adduce any evidence to support their belief. Instead, the problem is they cannot even define what the god is in which they claim to believe.