The arbitrariness argument

Recently came across the Euthyphro dilemma while reading about the Ontological argument. While the essential question it raises is of course pertinent, it does so in the context of morality, the relativistic nature of which in a logical debate raises problems.
Assume therefore that the concept of morality in the Euthyphro dilemma and in the arbitrariness argument is replaced with physically measurable (or mathematically computable/derivable) quantities - e.g. the cosmological constant, or the gravitational constant or the speed of light. A common theistic argument, the Fine Tuning Argument or Anthropic Principle is that for life to exist as it does, these constants must be precisely what they are. Let's assume this to be true. Let us also, for the sake of argument assume that the quantities were chosen by God to be so.
God being omniscient and omnipotent (two common assumptions in theology) could well have chosen other numbers for the above constants and still make the universe work in exactly the same way as it does right now. But he chose to go with the numbers we now measure or calculate to be true. In picking some numbers over the others, God is being arbitrary since no number is inherently better than others (42 of course, is an exception...).
Alternately, there are parallel universes out there with all possible combinations of these constants.
The first argument goes to show that arbitrariness is a necessary attribute of God if we believe in the concept of numbers. The second lacks physical proof and suffers from the same drawback - an omniscient being could well have interchanged these combinations amongst the other universes.
Since arbitrariness is generally thought of as an undesirable quality - one cannot 'trust' someone who is arbitrary with making 'correct' (quotations added to reflect fuzziness of the words and their inappropriateness in a logical discussion) decisions - we must conclude that our otherwise perfect God has arbitrariness as an undesirable adjoint attribute (alliteration!!).
The arbitrariness does not magically disappear even if one chooses to believe that there is a perfect reason somewhere for this set of numbers to be chosen unless, of course one considers numbers as being too illogical to figure in a logical argument.

One could alternately view God as the benevolent creator who let things proceed as they did. The constants then are not a product of design, but turn up that way just coincidentally and that we run into problems when we try to justify things by sticking probes and measuring quantities by assigning numbers to them - a limitation of man as it were. The essential problem with such a view is that it lies outside the domain of logical discussion and one enters the domain of belief, where, suddenly (and magically), our logical devices and faculties cease to function.