> Didn't the FLLF attack Palestinians?

Yes, and?

> Did it try to assassinate it's ambassador to the UK?

See below quotes. It also did try to assassinate US diplomat John Gunther Dean.

> Do you have other references to the theory that the reason Israel went to war with Lebanon was FLLF's operations that it blamed on Palestinians? Why would it go to war over people killing each other in Lebanon, it doesn't pass the smell test.

Not the sole reason in itself, rather a critical part of whipping political support.

[1]

> From his first day at the Defense Ministry, Sharon started planning the invasion of Lebanon. He developed what came to be known as the "big plan" for using Israel's military power to establish political hegemony in the Middle East. The first aim of Sharon's plan was to destroy the PLO's military infrastructure in Lebanon and to undermine it as a political organization. The second aim was to establish a new political order in Lebanon by helping Israel's Maronite friends, headed by Bashir Gemayel, to form a government that would proceed to sign a peace treaty with Israel. For this to be possible, it was necessary, third, to expel the Syrian forces fro Lebanon or at least seriously weaken their presence there. The destruction of the PLO would break the backbone of of Palestinian nationalism and facilitate the absorption of the West Bank into Greater Israel. The resulting influx of Palestinians from Lebanon into Jordan would eventually sweep away the Hashemite monarchy and transform the East Bank into a Palestinian state. Sharon reasoned that Jordan's conversion into a Palestinian state would end international pressures on Israel to withdraw from the West Bank.

> Sharon and Eytan, realizing there was no chance of persuading the cabinet to approve a large-scale operation in Lebanon, adopted a different tactic. They started presenting to the cabinet limited proposals for bombing PLO targets in Lebanon, expecting that the guerillas would retaliate by firing Katyusha rockets on Israel's northern settlements and that this would force the cabinet to approve more drastic measures. The idea was to implement Operation Big Pines in stages by manipulating enemy provocation and Israel's response. A number of confrontations took place in the cabinet as a result of these tactics. Ministers opposed to a war in Lebanon because they recognized where these proposals were intended to lead.

> Sharon himself displayed the same deviousness in his relations with the Reagan administration as he did in his relations with his cabinet colleagues. He fed the Americans selective information intended to prove that the PLO was making a mockery of the cease-fire agreement and to establish Israel's right to retaliate.

This coincides exactly with the FLLF terror campaign.

> On 3 June the casus belli that the hard-liners had been waiting for materialized. A group of Palestinian gunmen shot and greviously wounded Shlomo Argov, Israel's ambassador to London, outside the Dorchester Hotel.

> Mossad sources had intelligence to suggest that the attempt of Argov's life was intended to provoke an Israeli assault on Arafat's stronghold in Lebanon in order to break his power.

> Avraham Shalom, the head of the General Security Service, reported that the attack was most probably the work of the faction headed by Abu Nidal and suggested that Gideon Machanaimi, the prime minister's adviser on terrorism, elaborate on the nature of that organization. Machanaimi had hardly opened his mouth when Begin cut him off by saying, "They are all PLO."

[1] Shlaim - The Iron Wall, chapter "The Lebanese Quagmire"