Syrians and Lebanese people are basically the same people. The culture is very similar. Most are pretty secular and the religious are mostly not radicals. Qatar, Saudi and other countries are NOT "just the same" as Syria. They don't speak the same dialect and have an entirely different culture. Most in Syria OR Lebanon would be more comfortable in Europe than Arabia. Most would not really want to go further into the areas that are besieged by ISIS. I wouldn't! We keep hearing about them making insurgencies to Egypt, Arabia etc. Little tiny Bahrain and Qatar are NOT equipped for that kind of influx infrastructurally. The climate rather precludes just sticking them all in tent cities in the desert. They aren't Bedu, and the desert is not mildly desert-like. It is serious. Qatar has the money, but struggles to provide, even with all the money, water and sewage services to the current population.They are growing and trying to make more infrastructure, but they are likely overcome by the immigrant workers already there right now building the new infrastructure.
I wouldn't stop in Lebanon, I'd keep heading west. Not because I am greedy or a bad guy. Because ISIS has already rooted them from hearth and home. They want to feel safe again. They want their children to get an education. Look what happened to Palestinians. They want away from all the crazy political crap and I don't blame them.