Great post.  First, let me briefly reference the emboldened passages:

1)  Not only have I considered the possibility of an omnipotent entity that is "beyond logic," but I'm knowingly asserting my position in direct contrast to this possibility.  Furthermore, I held onto an agnostic position for a long time until further exploration suggested to me that the position is untenable.  I went from playing along with Roman Catholicism as a child, to atheism, to agnosticism, to a blend of agnosticism and East Asian religious philosophy (e.g. Buddhism, etc.), to complete open-mindedness, and now I've settled as a monistic theist.

2)  I appreciate that you tried to clarify and understand my position, because it's inaccurate.  I do not believe God can ever be revealed via the scientific method or any other line of inductive reasoning.  Even if we do not start with any presuppositions about God (because we shouldn't, else we put the cart before the horse), we already know right off the bat that inductive reasoning lacks the scope necessary to formulate absolute statements about reality at the highest possible level of generality.  Inductive reasoning fails because a presupposition about God would need to preclude any absolute statements made about such an entity.  In other words, we would need to somehow know absolutely what God is before finding any evidence to support that presupposition.  Obviously, this creates a huge problem, so we need a way to avoid the problem altogether.

Stemming from these two points, first we need to consider what is relevant to us, and perhaps the best way to identify what is relevant is to first identify which is irrelevant.  Specifically, things that are unreal or illogical are of no relevance to us because there is absolutely no possible way to make sense out of them.  If something were 'real' enough outside of reality so as to have an impact on it, then it would need to be inside reality.  Similarly, reality would be completely unintelligible if it weren't logical, and the fact that we all observe and interact with a stable Universe demonstrates that reality is inherently logical.

Now, let's focus on your phrasing when you talk about the possibility of an entity that is "beyond logic."  This is where things get fun. Really fun.  And really, really cool.

You could say that, in a sense, logic itself is beyond logic.  What I mean by this is that logic is holographic in structure.  I'm not sure how familiar you are with holograms, but if you take a piece of holographic film and you cut a corner from it, e.g. 25%, the result is not a corner that reveals 25% of the original image, but rather you have 100% of the image at 1/4 size.  Logic is similar. There are all kinds logical systems that vary according to scale, and although the sizes of these different systems vary, the logical properties governing all of them are the same.  

When we observe something, the logic and rules of observation (i.e. at a higher level) relate to the observed conditions (i.e. at a lower level) and allow us to make rational (remember, root word = ratio), statements about that relation.  Similarly, when we engage in metacognition, the logic and rules of metacognition (i.e. at a higher level) relate to various abstract objects of cognition (i.e. at a lower level) and allow us to make rational statements about that relation, too.  

The point I'm making is that we already have insight into how something 'beyond logic' works.  However, I would just clarify that it's not quite accurate to say 'beyond logic,' but rather it might be more accurate to say something like, "There exist logical systems of lower order that are necessitated by logical systems of higher order."  

If you're having trouble understanding what I mean by all of this, I'll refer you to an illustrative analogy I've used several times on this forum to demonstrate the point:  Imagine that we, as 3rd-dimensional beings, want to know what the 4th dimension is like. As 3D beings, we are limited by certain logical boundaries that define the 3rd dimension, so how can we know what the 4th dimension is like?   Well, what we can do is we can draw something like a tesseract, a 4th-dimensional object, on a 2nd-dimensional plane of paper.   But, similar to the problem of induction I spoke of earlier, how can we know that a tesseract is a sound model of a 4th-dimensional object?  Wouldn't that require that we invoke a presupposition of what a 4th-dimensional object is like before we've evidenced it?

As it turns out, when we draw a tesseract on a piece of paper, we are actually removing ourselves from the constraints of our 3rd-dimensional perspective, and instead we assume the perspective of a 5th-dimensional entity.  That is, we assume a 5d perspective and talk about the 4th dimension in the same way that we, as 3d beings, can fully explore and understand the logic of the 2nd-dimension.  All spatial dimensions are the same in their logical constructs (e.g. the 3rd-dimension can be described as the infinite sum of all 2d phenomena, the 4th-dimension can be described as the infinite sum of all 3d phenomena, etc.), but they vary according to scope.

To wrap up this post, notice how spatial dimensions are all logical constructs, but each successive dimension is *infinitely* greater than the previous one.  This should provide you with some insight into how we can gain insight into something (God?) that is infinitely greater than we or the logical system(s) we inhabit.

Edit:  Oh, I guess I should clarify what my actual position is.  My position is that God:Reality :: Man:Perceptions.  I believe it is accurate to say "man was created in God's image," and I think that we are all essentially gods...mini-gods.   I would venture so far as to say that, at the greatest possible scale, the interplay of consciousness and reality is God attempting to know himself and move towards self-actualization.