Generally speaking, Israel actually doesn't have that bad of an image of Lebanon and the Lebanese people. Of all the nations in the Middle East that they don't have relations with, Lebanon is the one people most strongly want to make peace with. The Israeli perception of Lebanon is of a Levantine nation that, like them, can balance European and Middle Eastern influences, and has beautiful landscapes and a great culture.
The problem that Israelis have is with their government, or more specifically the cancer that has infested it: The terrorist group Hezbollah. Given Hezbo's very blunt desire to annihilate Israel, that's really not a surprise. Israelis do recognize that they're a Lebanon-based group, but view them as an Iranian-backed leech so they don't really view it as all that Lebanese. And honestly, it's my impression that at least half the Lebanese themselves view Hezbollah quite negatively and too foreign as well, regardless of their stance on Israel.
I think Lebanese Christians will be quite surprised to find how friendly and warm Israelis can be, even if their backgrounds are known to them. Many Israelis view the friendlier Lebanese group to them to be the Christians, and recognize what group allied with them during the Lebanese Civil War. I know quite a few in the Lebanese Diaspora are also pro-Israel, interacting with some myself.
With Lebanese Jews, the answer is simple: They're Jews. Around 50% of Israel's Jewish population comes from communities that were established in nations hostile to them, so their Lebanese label doesn't scare them one bit. Jews recognize Jews as fellow Jews, it's as simple as that even for the Lebanese ones. However, when taking in the Lebanese label in specifically, most recognize the sad reality that the Lebanese Jewish community is effectively dead in Lebanon, with maybe 200 remaining in the nation (most of whom are elderly and stubborn to move).