I'm a theist. I totally agree that there are absolute truths. You're logic breaks down at step 2. If I don't agree with 1, your whole argument breaks down. In my experience debating philosophy throughout my education, seldom is there a Joker as in Heath Ledger's character. If a person fully believes that others don't count, at all, then the rule breaks down.
Take a counter example: homosexuals. Most societies, regardless of source, were anti-homosexual. This is historically accurate (modern times are uprooting them, but let's say we're in the 1950's).
1. I have the intuition that everyone ought to avoid homosexuality.
2. When I mean that everyone ought to avoid homosexuality, I am personally making a factual claim.
3. My intuition of 1 provides at least some evidence that 1 may be true.
4. If I have the intuition describe in 1, 1 is a factual claim, and my intuition of 1 provides at least some evidence for 1, then there is some evidence of an objective moral fact.
C. Therefore, there is some evidence for any objective moral fact.
We've now shown an implication that homosexuality is in fact immoral in a general sense. Are we ok with this? Why? Why not? Our ethics and morality, sans a center like an unmoved mover, can vacillate. Sometimes this is "good" (such as gay or interracial marriage [I am neither]) for me as it brings improved economic conditions that selfishly improve my lot directly or possibly, which I arbitrarily deem as good, from an agnostic world view. Sometimes it's "bad" (for example expelling Jews from the Spain since it took the engineers and literate people, which limited economic freedom and would have probably negatively impacted my Spanish life if my present personality and general thoughtful profession was true then).
As a quick aside for the postulate of cultural issues with homosexuality, http://news.trust.org//item/20140516162146-jipm9/ and https://en.wikipedia.org/wiki/Homosexuality_in_ancient_Rome which shows that Rome did not really have a great view of homosexual relationship. You didn't want to be the receiver. If you were, you were property.