I would say: "Sorry, I was saving a man's life!" But if you want to be honest, just say, "Sorry I'm late; I was having lunch." But, getting to a meeting late because you were having lunch? My goodness, that lunch must have been amazing!

"About being late" ("late" is not a verb, remember.) I agree with Rafael: "Sorry I'm late; I was having lunch" is exactly what you need. The only sentence which doesn't make sense is "Sorry, I am late because I had had my lunch." There's no need to use past perfect (had had) because you're not describing the past; you're explaining now.