I have experience losing my passport. --Sounds like you've lost it several times and you're good at what to do when that happens. The other two don't quite work. I would say (but hope never to have to): I have had the experience of losing my passport.

Native speakers just say, "I've lost my passport before" or "I once lost my passport." I am afraid none of your sentences sound natural to me at all.

What are you trying to say? If you want to say that you once lost your passport, you could say "I've had an experience of losing my passport"

The third sentence will not work. The only difference between sentence one and two is the preposition "of." Therefore, I would side with Irina's idea: "I have had an experience of losing my passport." Actually, as a native speaker myself, I would say, "I've lost my passport more than once." Or, "I have lost my passport before." I would lean away from using the word "experience," since losing your passport is already an experience, so you're being redundant. If you want to say you're good at it or you're disorganized or forgetful so you lose your passport from time to time, then you can say, "I have had several experiences of losing my passport." (Thank you for your focused question.)