No, Lebanese is not a “dialect” of Arabic
It would be an anachronism to assert that Italian is a dialect of Catalan, but safe to say that Italian comes from Latin. But when it comes to Lebanese (more generally NorthWestern Levantine), the “politically correct” Arabist-think-tank view is that is is derived from Arabic (Lebanese “dialect” of Arabic) to accommodate sensitivities — even linguists find arguments to violate the arrow of time to serve the interest of panArabism. In situations where there are similarities between a word used in Leb and Arabic, they insist it comes from Arabic not from a common root of both. (Most Lebanese are confused by diglossia as one is not supposed to write in the spoken language). Unlike Indo-European languages, Semitic languages have a criss-cross of roots and considerable areal diffusion to assert clean descendance, hence statements such as “A is a dialect of B” don’t have the certainty and neatness found elsewhere.
