Your primary conviction appears to be this: without some way to escape the arbitrariness of [allegedly] axiomatized moral systems, we have no way to ground the superiority of one moral system over another.
Perhaps you've labeled this question with the tag "moral skepticism" because you feel that such a position might exemplify a skeptical outlook. However, I believe that if you adopt a more thorough skepticism, you will find the need to abandon moral claims (and those fictitious 'moral axioms') altogether. One incarnation of this approach is emotivism, a variety of moral fictionalism.
In other words, there is no need to establish the superiority of one set of axioms over another (in the realm of ethics, at least) because all claims about good and bad behavior can be reduced without remainder to claims about emotion and preference.
Of course, if your main goal is to retain ethics as a legitimate field of study...or to retain God as the supreme rule-giver, then there certainly be some leftover after the reduction I've suggested above. However, doing this would amount to the unscientific attempt to bolster some previously-selected perspective. Additionally, we have no reason to accept ethics as legitimate and no reason to accept the reality of a Creator.
Consider for a moment that in this world there are no moral restrictions...and, of course, no hideous axioms from which philosophers and theologians are forced to infer hideous rules. There is fear, disgust, pain, happiness, and a bunch of emotions. There are also laws that differ from place to place and pieces of advice passed down from parent to child...from friend to friend. Some people approve of x, but not of y...and some of these people prefer to parse these feelings in terms of laws given from God or in terms of laws that were uncovered by way of mathematical deduction. The main goal of this sophistical trend appears to be this: if you can trick mankind into thinking that he must adhere to system A or system B, you can reduce those behaviors you consider offensive and encourage those behaviors you consider beneficial.
To answer in a different way. No, morality is not based on axioms but this does not imply that one system must be superior to another. The third option is that ethics itself is a towering pile of nonsense. A person, like myself, who subscribes to this line of thinking does not refrain from murder because it interferes with some immaterial law or because it implies some contradiction or because it originates from some superior ethical system; he decides not to murder because he finds the act to be a disgusting one that causes great harm.