Axiom: Everyone believes in something.
P1: I am skeptical of everything.
Conclusion: I believe that I am skeptical of everything
This conclusion cannot be drawn.
Even if we assume that "I am skeptical of everything" is synonymous with "I don't believe in anything", we cannot infer "I believe that I am skeptical of everything". The axiom ensures only that the individual making the assertion has some belief or other. It does not ensure that every statement within L is a belief.
I think you're right to assume that the axiom above would prevent someone from coherently asserting that they don't believe in anything. But to be honest, I'm a little unsure why you're attempting to rely on the inference (the conclusion), because we can obtain a contradiction prior to inferring anything if we assume that the two statements you consider equivalent are actually equivalent.
With unclear terms such as "skepticism", we'd be much better off talking about more precise phenomena such as withholding judgment, doubting the truth of some statement, or asserting that every statement is potentially falsifiable.
Also, I'm not sure how mainstream philosophy or logic would respond, but I think we have good reason to refrain from considering perceptual reports (There is a red car) or reports of outlook (I am skeptical of everything) as being on par with those things we normally think of when we refer to "beliefs": our attitudes toward some subjectively unverified assertion. If this is right, then it would be senseless to say "I believe that I am skeptical of everything"; you'd be, in a sense, unnecessarily tacking an attitude onto another attitude.