No reason justifies disbelief.

[The Grand Nudger]

I've been thinking about this intuition thing. It's totally bologna, lol. What is intuition in the first place? It's an unconscious assessment of empirical data, expressed as an innate, informed sense.  Without sense experience, there is no information, and so there couldn't be intuition.  Empiricism is the platform off of which intuition takes its leap. Sure, maybe its a "back of the house" assessment, like you said, but its still an assessment. That it feels different is irrelevant, and as you said, we use the same methods to distinguish between good/bad intuition and good/bad sense experience.  That's because they're essentially the same thing.  Giving something a different label doesn't magically transform it into something other than exactly what it is.

It's certainly possible that intuition is just another example of a subconscious form of information processing, and that this information is itself derived from sensory experience, but I think that a nominally rational intuitionist could object to the formulation above as proceeding from a question begging affirmation of empiricism, and even leverage the same closing remark - that calling every x empirical won't make it so any more than calling some empirical thing intuition would. 

I'd point out that you've given no reason to conclude that intuition can't be different from empirical observation and no reason that it would be impossible for intuition to provide knowledge.  You've given an opinion as to it's credibility/accuracy, and it's one that most (if not every) intuitionist would share with you.  The point of contention between the two positions simply isn't being addressed at all.  

How could I know that I exist without having a subjective experience first?

How could you have a subjective experience that didn't proceed from knowledge of self?  The existent self may be the axiom from which all statements of experience are made.  There is no "I see" without there first being an "I".  A skeptical response here could even be that you don't know that you exist, as you briefly mentioned before...but that you assume it.   With -or- without subjective experience, as those experiences can be in error, can be manufactured or populated with unrepresentative contents..or in the case of intuition above as we discussed it, may be happening entirely without your knowledge of them.  As innate knowledge would frame it, your self is not the kind of conclusion, but a necessary truth from which those other empirical observations are derived.  A favorite of innate knowledge theories are things like axioms and mathematics.  

Like before, with intuition, I could stab in the dark with examples but a more direct approach would be to ask if you can state with authenticity that you possess no innate knowledge whatsoever?  You seemed amenable to the idea of intuition, albeit leery with regards to it's accuracy, is it possible that innate knowledge presents a similar situation or circumstance?